Reduced Operator Algebras of

Trace-Preserving Quantum Automorphism Groups

Michael Brannan^{1}

Received: January 10, 2013

Communicated by Joachim Cuntz

Abstract. Let B be a finite dimensional C^{∗}-algebra equipped with its canonical trace induced by the regular representation of B on itself. In this paper, we study various properties of the trace-preserving quantum automorphism group 𝔾 of B. We prove that the discrete dual quantum group 𝔾̂ has the property of rapid decay, the reduced von Neumann algebra L^{∞}(𝔾) has the Haagerup property and is solid, and that L^{∞}(𝔾) is (in most cases) a prime type II_{1}-factor. As applications of these and other results, we deduce the metric approximation property, exactness, simplicity and uniqueness of trace for the reduced C^{∗}-algebra C_{r}(𝔾), and the existence of a multiplier-bounded approximate identity for the convolution algebra L^{1}(𝔾).

2000 Mathematics Subject Classification: Primary 46L65, 20G42; Secondary 46L54. Keywords and Phrases: Quantum automorphism groups, approximation properties, property of rapid decay, II_{1}-factor, solid von Neumann algebra, Temperley-Lieb algebra.

1 Introduction

Consider a finite set X_{n} consisting of n distinct points. An elementary and well known fact from group theory is that the automorphism group of C(X_{n}), the commutative algebra of complex valued functions over X_{n}, is precisely the permutation group S_{n} = Aut(X_{n}). In other words, the permutation group on n points is the universal object within the category of groups acting on C(X_{n}). In [40], Wang made the remarkable discovery that if one replaces the category of groups acting on C(X_{n}) by the category of compact quantum groups acting on C(X_{n}), then there still exists a universal object within this new category, called the quantum permutation group S_{n}^{+}. S_{n}^{+} is a C^{∗}-algebraic compact quantum group which contains the classical permutation group S_{n} as a quantum subgroup. Moreover, the inclusion S_{n} ⊆ S_{n}^{+} is proper when n ≥ 4. That is, finite sets with at least four elements admit genuinely “quantum permutations”. More generally, given a (non-commutative) finite dimensional C^{∗}-algebra B, one can ask about the existence of a universal object within the category of compact quantum groups acting on B. This question was also considered by Wang [40], who showed that such an object exists provided we fix a faithful state ψ : B → ℂ and restrict to the subcategory of compact quantum groups acting on B which preserve the state ψ. The resulting object is called the quantum automorphism group of the pair (B, ψ), and is denoted by 𝔾_{aut}(B, ψ). When B = C(X_{n}) and ψ is the uniform probability measure on X_{n}, we recover the quantum permutation group S_{n}^{+}. See Example 3.4. Since the quantum groups 𝔾_{aut}(B, ψ) generally depend a great deal on the initial choice of state ψ, a natural first step in attempting to develop a structure theory for these objects is to restrict attention to certain “canonical” choices of states on B (which should ideally correspond to uniform probability measures in the commutative case). In [3, 4], Banica showed that one very natural choice of state ψ : B → ℂ, which ensures that the generators of the category of finite dimensional representations of 𝔾 = 𝔾_{aut}(B, ψ) satisfy the fewest relations (or are as “free” as possible), is to take ψ to be the restriction to B of the unique trace on L(B), where B ↪ L(B) via the left regular representation. In particular, with the above choice of ψ (or more generally by taking ψ to be a δ-form on B – see Definition 3.5), Banica showed that when dim B ≥ 4, the finite dimensional representation theory of 𝔾 admits a nice description using Temperley-Lieb algebras. Using this link with Temperley-Lieb algebras, he obtained a complete description of the fusion rules for the irreducible unitary representations of 𝔾 and from this deduced many interesting structure results, including the fact that 𝔾 is co-amenable if and only if dim B ≤ 4. The goal of this paper is to investigate the operator algebraic structure of the quantum groups 𝔾 = 𝔾_{aut}(B, ψ), where ψ : B → ℂ is the canonical trace described above. Our focus will be on properties of the reduced C^{∗}- algebras C_{r}(𝔾) and reduced von Neumann algebras L^{∞}(𝔾) = C_{r}(𝔾)′′, as well as the quantum convolution algebras L^{1}(𝔾) = L^{∞}(𝔾)_{∗}. Note that in the coamenable (i.e., dim B ≤ 4) case, the structure of 𝔾 is already well understood. If dim B = n ≤ 3, then B = C(X_{n}) and 𝔾 = S_{n}. If dim B = 4, then either B = M_{2}(ℂ) or C(X_{4}). In the first case, 𝔾 turns out to be isomorphic to SO(3), the classical ∗-automorphism group of M_{2}(ℂ) (see [3]). In the second case, 𝔾 = S_{4}^{+}, which has been extensively studied in [5] and [7]. In particular, Banica and Collins [7] have obtained a concrete model for the reduced (= full, due to co-amenability) C^{∗}-algebra C_{r}(S_{4}^{+}) in the form of an explicit embedding C_{r}(S_{4}^{+}) ↪ M_{4}(ℂ) ⊗ C(SU(2)). In the non-co-amenable (i.e., dim B ≥ 5) regime, much less is known about the structure of 𝔾 and its operator algebras. Among the known results, we note that De Rijdt and Vander Vennet [20] have obtained a detailed description of the probabilistic boundaries associated to the discrete dual quantum group 𝔾̂. In addition, combinatorial “Weingarten” formulas for the Haar state on the quantum permutation groups S_{n}^{+} were obtained in [6]. The large n asymptotics of these formulas have found many interesting applications in free probability theory. See for example [8, 9, 28]. Perhaps most striking of these applications is Köstler and Speicher’s free analogue of de Finetti’s theorem [28], which asserts that an infinite sequence of random variables in a W^{∗}-probability space is identically distributed and free with amalgamation if and only if its joint ∗- distribution is invariant under the action of the quantum permutation groups {S_{n}^{+}}_{n≥1}. Returning now to a fixed quantum automorphism group 𝔾 = 𝔾_{aut}(B, ψ) with dim B ≥ 5, basic results in co-amenability theory imply that C_{r}(𝔾) is nonnuclear, L^{∞}(𝔾) is non-injective and L^{1}(𝔾) is a Banach algebra which fails to have a bounded approximate identity. However, based on the above connections between S_{n}^{+} and free independence, as well as some compelling evidence provided by work of Vaes and Vergnioux [36] on free orthogonal quantum groups, more is conjectured to be true. Namely, it is expected that L^{∞}(𝔾) is a full, prime type II_{1}-factor and C_{r}(𝔾) is a simple C^{∗}-algebra with unique trace when dim B ≥ 5 (see [7], Introduction). In this paper we show that if dim B ≥ 8, then L^{∞}(𝔾) is a full type II_{1}-factor (Theorem 5.1) and C_{r}(𝔾) is simple with unique trace (Corollary 5.12) – verifying the above conjecture in all but six cases. We also prove that L^{∞}(𝔾) is a solid von Neumann algebra (Corollary 5.14): the relative commutant of any diffuse injective von Neumann subalgebra of L^{∞}(𝔾) is injective. Combining this with the above factoriality result, we conclude that L^{∞}(𝔾) is a prime factor as soon as dim B ≥ 8. The solidity of L^{∞}(𝔾) is established by proving that L^{∞}(𝔾) has property (AO)^{+} (Theorem 5.13), which implies solidity by a celebrated result of Ozawa [31]. See Section 5.4 for the relevant details and definitions. Our proof of property (AO)^{+} follows very closely the method of Vergnioux [37], where property (AO)^{+} was established for universal discrete quantum groups. In addition to considering the algebraic structure of L^{∞}(𝔾) and C_{r}(𝔾), we also investigate various approximation properties for these operator algebras and the convolution algebra L^{1}(𝔾). We prove that when dim B ≥ 5, L^{∞}(𝔾) has the Haagerup property (Theorem 4.2), C_{r}(𝔾) (respectively L^{∞}(𝔾)) has the (respectively weak^{∗}) metric approximation property, and that L^{1}(𝔾) admits a central approximate identity which is contractive in the multiplier-norm (Theorem 4.15). To obtain these approximation properties, we first construct a suitable family of L^{2}-compact completely positive multipliers on L^{∞}(𝔾), establishing the Haagerup property. We then prove that the discrete dual quantum group 𝔾̂ has the property of rapid decay, which amounts to proving a type of Haagerup inequality for L^{∞}(𝔾) (Theorem 4.10). The norm control provided by the property of rapid decay for 𝔾̂ allows us to truncate our completely positive multipliers down to finite rank contractions in such a way as to yield the (weak^{∗}) metric approximation property. (We also use the property of rapid decay in the proof of simplicity for C_{r}(𝔾)). The existence of the claimed approximate identity for L^{1}(𝔾) then follows by a standard duality argument. We also observe that C_{r}(𝔾) is an exact C^{∗}-algebra (Corollary 4.17), by piecing together some results on monoidal equivalence for compact quantum groups from [20] and [36]. The remaining parts of this paper are organized as follows: Section 2 fixes some notation and contains the basic facts on compact and discrete quantum groups that will be needed. In Section 3 we recall the definition of the quantum automorphism groups 𝔾 = 𝔾_{aut}(B, ψ), δ-forms and describe the representation theory of 𝔾 when ψ is a δ-form. Section 4 deals with approximation properties for the reduced operator algebras of 𝔾 (assuming ψ is the canonical trace on B) and the property of rapid decay for 𝔾̂. In Section 5 we prove our factoriality, fullness and simplicity results for the reduced operator algebras of 𝔾. We end Section 5 by establishing property (AO)^{+} for L^{∞}(𝔾), deducing solidity and primeness results from this. The final section (Section 6) is an appendix containing a proof of Theorem 5.4, which is a technical result required in our study of factoriality and fullness for L^{∞}(𝔾).

1Research partially supported by an NSERC Postdoctoral Fellowship.

Documenta Mathematica 18 (2013) 1349–1402

Acknowledgements

The author wishes to thank T. Banica, B. Collins, P. Fima, M. Junge, J. A. Mingo, R. Speicher and R. Vergnioux for useful conversations and comments at various stages of this project.

2 Preliminaries

2.1 Notation

For a Hilbert space H, we take its inner product to be conjugate-linear in the first variable and write B(H) for the C^{∗}-algebra of bounded linear operators on H. Given ξ, η ∈ H, we write ω_{η,ξ} ∈ B(H)_{∗} for the σ-weakly continuous linear functional T ↦ ⟨η|T ξ⟩. For tensor products, we write ⊗ for the minimal tensor product of C^{∗}-algebras or Hilbertian tensor product of Hilbert spaces, \overline{⊗} for the spatial tensor product of von Neumann algebras and ⊗_{alg} for the algebraic tensor product. If X is an operator space, we write CB(X) for the algebra of completely bounded linear maps on X. We take the natural numbers ℕ to include 0 and for m ≥ 1 we write [m] := {1, 2, …, m}. We also use standard leg numbering notation for operators on multiple tensor products. For example, if H_{1}, H_{2}, H_{3} are Hilbert spaces and T ∈ B(H_{1} ⊗ H_{2}), then T_{12} = T ⊗id_{H_{3}} ∈ B(H_{1}⊗H_{2}⊗H_{3}), T_{13} = (id⊗σ^{∗})T_{12}(id⊗σ) ∈ B(H_{1}⊗H_{3}⊗H_{2}) where σ : H_{3}⊗H_{2} → H_{2}⊗H_{3} is the tensor flip map and T_{23} is defined similarly.

2.2 Compact and Discrete Quantum Groups

We present here a very brief summary of the basic facts on compact and discrete quantum groups that will be needed for this paper. Our main reference for this will be [43] and the excellent book [35].

Definition 2.1. A compact quantum group is a pair 𝔾 = (A, Δ), where A is a unital C^{∗}-algebra and Δ : A → A ⊗ A is a unital ∗-homomorphism (called a coproduct) satisfying the coassociativity condition

(id_{A} ⊗ Δ) ∘ Δ = (Δ ⊗ id_{A}) ∘ Δ

and the cancellation property

\overline{span}\{Δ(A)(1_{A} ⊗ A)\} = A ⊗ A = \overline{span}\{Δ(A)(A ⊗ 1_{A})\}.

Every compact quantum group 𝔾 = (A, Δ) admits a Haar state, which is a state h : A → ℂ determined uniquely by the following bi-invariance condition with respect to the coproduct:

(h ⊗ id_{A})Δ(a) = (id_{A} ⊗ h)Δ(a) = h(a)1_{A} (a ∈ A).

Let L^{2}(𝔾) := L^{2}(A, h) denote the GNS Hilbert space associated to the state h and let λ : A → B(L^{2}(𝔾)) the left regular representation. That is, L^{2}(𝔾) is Hilbert space obtained from A by separation and completion with respect to the sesquilinear form ⟨a|b⟩_{h} = h(a^{∗}b), and λ is given by λ(a)Λ_{h}(b) = Λ_{h}(ab), where a, b ∈ A and Λ_{h} : A → L^{2}(𝔾) is the GNS map. The reduced C^{∗}-algebra and reduced von Neumann algebra of 𝔾 are the concrete operator algebras acting on L^{2}(𝔾) given by

C_{r}(𝔾) := λ(A) ⊆ B(L^{2}(𝔾))  and  L^{∞}(𝔾) := C_{r}(𝔾)′′,

respectively. Observe that the Δ-invariance of the Haar state h implies that there exists a normal faithful ∗-homomorphism Δ_{r} : L^{∞}(𝔾) → L^{∞}(𝔾)\overline{⊗}L^{∞}(𝔾) given by Δ_{r} ∘ λ = (λ ⊗ λ) ∘ Δ, turning (C_{r}(𝔾), Δ_{r}) into a compact quantum group (the reduced version of 𝔾). The pre-adjoint of Δ_{r} induces a completely contractive Banach algebra structure on the predual L^{1}(𝔾) := L^{∞}(𝔾)_{∗}. L^{1}(𝔾) is called the convolution algebra of 𝔾. 𝔾 is said to be of Kac-type if the Haar state h is tracial. In this case, L^{∞}(𝔾) is a finite von Neumann algebra with faithful Haar trace h. Let H be a finite dimensional Hilbert space. A representation of 𝔾 on H is an element U ∈ B(H) ⊗ A such that

(id ⊗ Δ)U = U_{12}U_{13}.  (2.1)

By fixing an orthonormal basis {e_{i}}_{i=1}^{d} for H and writing U = [u_{ij}] ∈ M_{d}(A) relative to this basis, (2.1) is equivalent to requiring that Δ(u_{ij}) = ∑_{k=1}^{d} u_{ik} ⊗ u_{kj} for 1 ≤ i, j ≤ d. U is called a unitary representation if U is, in addition, a unitary element of M_{d}(A). If U^{1} = [u_{i(1)j(1)}^{1}] ∈ M_{d(1)}(A) and U^{2} = [u_{i(2)j(2)}^{2}] ∈ M_{d(2)}(A) are (unitary) representations, then their tensor product U^{1} ⊠ U^{2} = [u_{i(1)j(1)}^{1}u_{i(2)j(2)}^{2}] ∈ M_{d(1)d(2)}(A) and direct sum U^{1} ⊕ U^{2} ∈ M_{d(1)+d(2)}(A) are also (unitary) representations. The vector space

Mor(U^{1}, U^{2}) = {T ∈ B(H_{1}, H_{2}) | (T ⊗ 1_{A})U^{1} = U^{2}(T ⊗ 1_{A})},

is called the space of morphisms (or intertwiners) between U^{1} and U^{2}. U^{1} and U^{2} are called (unitarily) equivalent if there exists an invertible (unitary) operator T ∈ Mor(U^{1}, U^{2}), and in this case we write U^{1} ≅ U^{2}. If U = [u_{ij}] ∈ M_{d}(A) is a representation, then so is \overline{U} = [u_{ij}^{∗}]. The representation \overline{U} is called the conjugate of U. Note that the passage from U to its conjugate \overline{U} does not preserve unitarity in general. A representation U is called irreducible if Mor(U, U) = ℂid. Note that U is irreducible if and only if \overline{U} is irreducible. The following is a fundamental result in the theory of compact quantum groups.

Theorem 2.2. Every irreducible representation of a compact quantum group 𝔾 = (A, Δ) is finite dimensional and equivalent to a unitary one if it is invertible. Moreover, every unitary representation is unitarily equivalent to a direct sum of irreducibles.

Let {U^{α} = [u_{ij}^{α}]_{1≤i,j≤d_{α}} : α ∈ Irr(𝔾)} be a maximal family of pairwise inequivalent finite-dimensional irreducible unitary representations of 𝔾 = (A, Δ), with 0 ∈ Irr(𝔾) denoting the index corresponding to the trivial representation 1_{A} ∈ A and U\overline{α} denoting the representative of the equivalence class of \overline{U^{α}}. If A ⊂ A denotes the linear span of {u_{ij}^{α} : 1 ≤ i, j ≤ d_{α}, α ∈ Irr(𝔾)}, then A is a Hopf ∗-algebra on which h is faithful, A is norm dense in A, and the set {u_{ij}^{α} : 1 ≤ i, j ≤ d_{α}, α ∈ Irr(𝔾)} is a linear basis for A. The coproduct of A arises as the restriction of the coproduct of A, so in particular Δ(A) ⊆ A⊗_{alg}A. The coinverse κ : A → A is the antihomomorphism given by κ(u_{ij}^{α}) = (u_{ji}^{α})^{∗}, and the counit ϵ : A → ℂ is the ∗-character given by ϵ(u_{ij}^{α}) = δ_{ij}. The Hopf ∗-algebra A is unique in the sense that if B ⊆ A is any other dense Hopf ∗-subalgebra, then B = A (see [10, Theorem 5.1]). For each α ∈ Irr(𝔾), fix an isometric morphism t_{α} ∈ Mor(1, U^{α} ⊠ U\overline{α}) (which exists and is unique up to scalar multiplication by 𝕋). Let j_{α} : H_{α} → H\overline{α} be the invertible conjugate-linear map defined by j_{α}ξ = (ξ^{∗} ⊗ id)t_{α}, (ξ ∈ H_{α}). Renormalize j_{α} so that the positive operator Q_{α} := j_{α}^{∗}j_{α} > 0 satisfies Tr(Q_{α}) = Tr(Q_{α}^{−1}) and j\overline{α}j_{α} = ±id_{H_{α}}. The operators \{Q_{α}\}_{α∈Irr(𝔾)} can then be used to describe the Schur orthogonality relations for the matrix elements of irreducible unitary representations of 𝔾, which are given by

h((u_{ij}^{α})^{∗}u_{kl}^{β}) = \frac{δ_{αβ}δ_{jl}(Q_{α}^{−1})_{ki}}{TrQ_{α}} and h(u_{ij}^{α}(u_{kl}^{β})^{∗}) = \frac{δ_{αβ}δ_{ik}(Q_{α})_{lj}}{TrQ_{α}} , (2.2)

Documenta Mathematica 18 (2013) 1349–1402

where α, β ∈ Irr(𝔾) and 1 ≤ i, j ≤ d_{α}, 1 ≤ k, l ≤ d_{β}. See [43] for details. In particular, setting L_{α}^{2}(𝔾) = span{Λ_{h}(u_{ij}^{α}) : 1 ≤ i, j ≤ d_{α}}, we obtain the L^{2}(𝔾)-decomposition

L^{2}(𝔾) = ⨁_{α∈Irr(𝔾)} L_{α}^{2}(𝔾) ≅ ⨁_{α∈Irr(𝔾)} H_{α} ⊗ H\overline{α},  (2.3)

where the second isomorphism is given on each direct summand L_{α}^{2}(𝔾) by

Λ_{h}((ω_{η,ξ} ⊗ id_{A})U^{α}) ↦ ξ ⊗ (1 ⊗ η^{∗})t\overline{α}  (α ∈ Irr(𝔾), ξ, η ∈ H_{α}). (2.4)

Note that when 𝔾 is of Kac-type, then Q_{α} = id_{H_{α}} for each α ∈ Irr(𝔾) and {d_{α}^{1/2}Λ_{h}(u_{ij}^{α}) : 1 ≤ i, j ≤ d_{α}} is then an orthonormal basis for L_{α}^{2}(𝔾). For future reference, we will always write P_{α} for the orthogonal projection from L^{2}(𝔾) onto L_{α}^{2}(𝔾) ≅ H_{α} ⊗ H\overline{α}. Denote by C_{u}(𝔾) the universal enveloping C^{∗}-algebra of the Hopf ∗-algebra A and by π_{u} : A → C_{u}(𝔾) the universal representation. Then there is a ∗- homomorphism Δ_{u} : C_{u}(𝔾) → C_{u}(𝔾) ⊗ C_{u}(𝔾) defined by Δ_{u} ∘ π_{u} = (π_{u} ⊗ π_{u}) ∘ Δ, making (C_{u}(𝔾), Δ_{u}) a compact quantum group (the universal version of 𝔾). In the universal setting, we write C_{alg}(𝔾) for the canonical dense Hopf ∗-subalgebra π_{u}(A) ⊆ C_{u}(𝔾). We say that 𝔾 is co-amenable if the regular representation λ : C_{u}(𝔾) → C_{r}(𝔾) is an isomorphism. The compact quantum groups encountered in this paper will all appear canonically in universal form. I.e., given by 𝔾 = (A, Δ) where A = C_{u}(𝔾) and Δ = Δ_{u}. Following [32], we view a discrete quantum group as the dual of a compact quantum group: Let 𝔾 = (A, Δ) be a compact quantum group and let {U^{α}}_{α∈Irr(𝔾)} (with U^{α} ∈ B(H_{α})⊗A) be a maximal family of pairwise inequivalent irreducible unitary representations of 𝔾. Define

ℂ[𝔾̂] = ⨁_{α∈Irr(𝔾)} B(H_{α}), c_{0}(𝔾̂) = c_{0} − ⨁_{α∈Irr(𝔾)} B(H_{α}),

ℓ^{∞}(𝔾̂) = ∏_{α∈Irr(𝔾)} B(H_{α}),

and let P̂_{α} ∈ ℓ^{∞}(𝔾̂) denote the minimal central projection corresponding to the factor B(H_{α}) ∈ ℓ^{∞}(𝔾̂). The unitary operator

𝕍 = ⨁_{α∈Irr(𝔾)} U^{α} ∈ M(c_{0}(𝔾̂) ⊗ A),  (2.5)

where M(B) denotes the multiplier algebra of a C^{∗}-algebra B, implements the duality between 𝔾 and 𝔾̂. There is a dual coproduct Δ̂ : ℓ^{∞}(𝔾̂) → ℓ^{∞}(𝔾̂)\overline{⊗}ℓ^{∞}(𝔾̂) determined by

(Δ̂ ⊗ id_{A})𝕍 = 𝕍_{13}𝕍_{23},

or equivalently by the relation Δ̂(x)S = Sx for each x ∈ ℓ^{∞}(𝔾̂), S ∈ Mor(U^{α}, U^{β} ⊠ U^{γ}). The left and right invariant Haar weights on ℓ^{∞}(𝔾̂), ĥ_{L} and ĥ_{R}, are given by

ĥ_{L}(x) = ∑_{α∈Irr(𝔾)} m_{α}Tr(Q_{α}P̂_{α}x),  ĥ_{R}(x) = ∑_{α∈Irr(𝔾)} m_{α}Tr(Q_{α}^{−1}P̂_{α}x) (2.6)

for each x ∈ ℂ[𝔾̂], where m_{α} = Tr(Q_{α}) = Tr(Q_{α}^{−1}) denotes the quantum dimension of H_{α}. Finally, we define the positive invertible unbounded multiplier Q = ∑_{α∈Irr(𝔾)} P̂_{α}Q_{α} of c_{0}(𝔾̂). Note that Q satisfies the relation Δ̂(Q) = Q⊗Q and plays the role of the Haar modular function on a locally compact group.

3 Quantum Automorphism Groups of Finite Dimensional C^{∗}- algebras.

We now turn to the central objects of this paper – quantum automorphism groups of finite dimensional C^{∗}-algebras. To define these objects, we first recall the notion of an action of a compact quantum group on a unital C^{∗}-algebra.

Definition 3.1. Let B be a unital C^{∗}-algebra and 𝔾 = (A, Δ) a compact quantum group.

1. A right action of 𝔾 on B is a unital ∗-homomorphism α : B → B ⊗ A satisfying

(α ⊗ id)α = (id ⊗ Δ)α  and  \overline{span}\{α(B)(1 ⊗ A)\} = B ⊗ A.

2. If ψ : B → ℂ is a positive linear functional on B, a right action α of 𝔾 on the pair (B, ψ) is a right action of 𝔾 on B which is also ψ-invariant. That is,

(ψ ⊗ id_{A})α(b) = ψ(b)1_{A}, (b ∈ B).

Definition 3.2. ([40, 3, 4]) Let (B, ψ) be a finite dimensional C^{∗}-algebra equipped with a faithful state ψ. The quantum automorphism group of (B, ψ), denoted by 𝔾 = 𝔾_{aut}(B, ψ), is the universal quantum group defined by a right action of 𝔾 on (B, ψ), say α : B → B ⊗ C_{u}(𝔾), with the following properties:

1. C_{u}(𝔾) is defined as the universal C^{∗}-algebra generated by

{(ω ⊗ id)α(a) : ω ∈ B^{∗}, a ∈ B}.

2. The action α is universal in the sense that if β : B → B ⊗ A′ is an action of another compact quantum group 𝔾′ = (A′, Δ′) on (B, ψ), then there exists a unital ∗-homomorphism π : C_{u}(𝔾) → A′ such that β = (id_{B} ⊗ π)α.

Remark 3.3. As was shown in [40, 3, 4], it is possible to give a more “concrete” description of the quantum group 𝔾 = 𝔾_{aut}(B, ψ) in terms of generators and relations. Let n = dim B and fix an orthonormal basis {b_{i}}_{i=1}^{n} for B relative to the inner product ⟨·|·⟩_{ψ} induced by ψ. Let m : B⊗B → B be the multiplication map and ν : ℂ → B the unit map. Then C_{u}(𝔾) is the universal C^{∗}-algebra with n^{2} generators {u_{ij} : 1 ≤ i, j ≤ n} subject to the relations which make

U = [u_{ij}] ∈ M_{n}(C_{u}(𝔾)) unitary, m ∈ Mor(U^{⊠2}, U) and ν ∈ Mor(1, U). (3.1)

The universal property of C_{u}(𝔾) then allows us to define a coproduct Δ : C_{u}(𝔾) → C_{u}(𝔾) ⊗ C_{u}(𝔾) by the formula

Δ(u_{ij}) = ∑_{k=1}^{n} u_{ik} ⊗ u_{kj} (1 ≤ i, j ≤ n),

which turns (C_{u}(𝔾), Δ) into a compact quantum group with fundamental representation U acting on the Hilbert space H = (B, ⟨·|·⟩_{ψ}). (Recall that a fundamental representation of a compact quantum group 𝔾 = (A, Δ) is a representation U whose matrix elements generate A as a C^{∗}-algebra.) The universal action α of 𝔾 on (B, ψ) is uniquely determined by α(b_{i}) = ∑_{j=1}^{n} b_{j} ⊗ u_{ji}, (1 ≤ i ≤ n). Example 3.4. Consider the simplest situation where X_{n} = {1, …, n}, B = C(X_{n}) and ψ is the uniform probability measure on X_{n}. Taking as an orthogonal basis the n standard projections in C(X_{n}), it can be shown using Remark 3.3 that C_{u}(𝔾) is the universal C^{∗}-algebra generated by n^{2} projections {u_{ij} : 1 ≤ i, j ≤ n} with the property that the rows and columns of the matrix U = [u_{ij}] form partitions of unity. (An n × n matrix U over a C^{∗}-algebra A satisfying the above properties is called a magic unitary.) On the other hand, C_{u}(S_{n}^{+}), the universal C^{∗}-algebra associated to the quantum permutation group S_{n}^{+}, admits the same description by [40], Theorem 3.1. Therefore 𝔾_{aut}(C(X_{n}), ψ) = S_{n}^{+}. As mentioned in the introduction, this paper deals almost exclusively with finite dimensional C^{∗}-algebras equipped with certain canonical tracial states, called tracial δ-forms [3, 4].

Definition 3.5. Let B be a finite dimensional C^{∗}-algebra equipped with a faithful state ψ : B → ℂ and let δ > 0. We call ψ a δ-form if for the inner products on B and B ⊗ B implemented by ψ and ψ ⊗ ψ, respectively, the multiplication map m : B ⊗ B → B satisfies mm^{∗} = δ^{2}id_{B}.

The reason for considering δ-forms is two-fold. On the one hand, the class of δ-forms yields many of the interesting concrete examples of quantum automorphism groups (including quantum permutation groups – see Example 3.7 below). On the other hand, it turns out that the concrete monoidal W^{∗}- category of finite dimensional unitary representations of 𝔾_{aut}(B, ψ) admits a very tractable description in this setting (see Section 3.1). For a brief discussion of the situation where ψ is not a δ-form, we refer the reader to the concluding remark at the end of the paper in Section 5.5.

Remark 3.6. Let ψ be a δ-form on a finite dimensional C^{∗}-algebra B. One can readily verify (see for example [4], Theorem 1) that in addition to usual relations

mm^{∗} = δ^{2}id_{B}, ν^{∗}ν = 1, m(m ⊗ id_{B}) = m(id_{B} ⊗ m),
m(id_{B} ⊗ ν) = m(ν ⊗ id_{B}) = id_{B}

satisfied by the multiplication map m : B⊗B → B and the unit map ν : ℂ → B, the additional useful relation

m^{∗}m = (m ⊗ id_{B})(id_{B} ⊗ m^{∗}),

also holds.

We now consider some examples.

Example 3.7. 1. If B = C(X_{n}) where X_{n} = {1, 2, …, n} and ψ is the uniform probability measure on X_{n}, then ψ is a δ-form with δ^{2} = dim B = n. Note that by Example 3.4, 𝔾_{aut}(B, ψ) = S_{n}^{+}.

2. If B = M_{n}(ℂ) and ψ = Tr(Q·) with Q > 0, then ψ is a δ-form with δ^{2} = Tr(Q^{−1}).

3. If (B, ψ) is arbitrary, then B decomposes as the direct sum B = ⨁_{i=1}^{k} M_{n_{i}}(ℂ) and ψ can be written as ψ = ⨁_{i=1}^{k} Tr(Q_{i}·) with Q_{i} > 0. Then ψ is a δ-form if and only if Tr(Q_{i}^{−1}) = δ^{2} for 1 ≤ i ≤ k. See [4].

Using the notation from Example 3.7(3) and the inequality dim B ≤ ∑_{1≤i≤k} Tr(Q_{i})Tr(Q_{i}^{−1}), it follows that δ^{2} ≥ dim B for any δ-form ψ. If ψ is moreover a trace, then δ^{2} = dim B and ψ is precisely the restriction to B of the unique trace on L(B), described in Section 1. In this case, we call ψ the δ-trace or tracial δ-form on B. Note that when ψ is a trace, 𝔾_{aut}(B, ψ) is a compact quantum group of Kac-type [40]. We remind the reader that for n = dim B ≤ 3, 𝔾 = 𝔾_{aut}(B, ψ) is just the usual permutation group S_{n} [40], so for the remainder of the paper we will assume that dim B ≥ 4.

3.1 Representation Theory and the 2-cabled Temperley-Lieb Category

Let δ ≥ 2 and ψ be a δ-form on a finite dimensional C^{∗}-algebra B. In this section we review the description of the representation theory of 𝔾 = 𝔾_{aut}(B, ψ) obtained by Banica in [3, 4]. In the following paragraphs it will be convenient to use the language of (concrete) monoidal W^{∗}-categories. We refer to [42] and [23] for the appropriate definitions.

In [4] (see [3] for the tracial case) it is shown that the fundamental representation U of 𝔾 is equivalent to \overline{U}, and therefore the concrete monoidal W^{∗}-category R of all finite dimensional unitary representations of 𝔾 is the completion of the concrete monoidal W^{∗}-category

R_{0} = {{U^{⊠k}}_{k∈ℕ}, {Mor(U^{⊠k}, U^{⊠l})}_{k,l∈ℕ}},

whose objects are the tensor powers U^{⊠k} of U and whose morphisms are the associated spaces of intertwiners. Moreover, R_{0} is generated as a monoidal W^{∗}-category by the morphisms m ∈ Mor(U^{⊠2}, U), ν ∈ Mor(1, U) and id_{B} ∈ Mor(U, U). Consider the universal graded C^{∗}-algebra (TL_{k,l}(δ))_{(k,l)∈ℕ×ℕ} given by TL_{k,l}(δ) = {0} if k−l is odd, and generated by elements t(k, l) ∈ TL_{k+l,k+l+2}(δ) with the following relations (where 1_{n} denotes the unit of the C^{∗}-algebra TL_{n}(δ) := TL_{n,n}(δ)):

t(k, l)^{∗}t(k, l) = 1_{k+l}
t(k, l + 1)^{∗}t(k + 1, l) = δ^{−1}1_{k+l+1}
t(r, k + l + 2)t(r + k, l) = t(r + k + 2, l)t(r, k + l)
t(r, k + l + 2)^{∗}t(r + k + 2, l) = t(r + k, l)t(r, k + l)^{∗}.

It is well known that each (necessarily finite dimensional!) vector space TL_{k,l}(δ) has the following planar-diagrammatic interpretation: TL_{k,l}(δ) is the ℂ-vector space spanned by a basis of (k, l)-Temperley-Lieb diagrams {D_{π}}_{π∈NC_{2}(k+l)}. (That is, diagrams D_{π} consisting of two parallel rows of points – k points on the bottom row and l points on the top row – which are connected by a non-crossing pairing π ∈ NC_{2}(k + l)). We refer to the book [27] for details on the spaces T L_{k,l}(δ). Note, however, that our choice of notation and terminology differs slightly from theirs. In terms of diagrams, the generators t(k, l) ∈ TL_{k+l,k+l+2}(δ) are given by the (scaled) diagrams

t(k, l) = δ^{−1/2} \underbrace{| | ⋯ |}_{k dots} ⋃ \underbrace{| ⋯ | |}_{l dots}, t(k, l)^{∗} = δ^{−1/2} \underbrace{| | ⋯ |}_{k dots} ⋂ \underbrace{| ⋯ | |}_{l dots} .

The composition D_{π}D_{σ} ∈ TL_{k,l}(δ) of diagrams D_{π} ∈ TL_{s,l}(δ), D_{σ} ∈ TL_{k,s}(δ) is obtained by the following procedure. First stack the diagram D_{π} on top of D_{σ}, connecting the bottom row of s points on D_{π} to the top row of s points on D_{σ}. The result is a new planar diagram, which may have a certain number c of internal loops. By removing these loops, we obtain a new diagram D_{ρ} ∈ TL_{k,l}(δ). The product D_{π}D_{σ} ∈ TL_{k,l}(δ) is then defined to be δ^{c}D_{ρ}. The involution D_{π} ↦ D_{π}^{∗} is just the conjugate linear extension of the operation of turning diagrams upside-down. The collection TL(δ) = {ℕ, {TL_{k,l}(δ)}_{k,l∈ℕ}} has a natural structure as a monoidal W^{∗}-category and is called the Temperley-Lieb category (with parameter δ). Related to this is the 2-cabled Temperley-Lieb category TL^{2}(δ) = {ℕ, {TL_{k,l}^{2}(δ)}_{k,l∈ℕ}}, where TL_{k,l}^{2}(δ) := T L_{2k,2l}(δ) for each k, l ∈ ℕ. A fundamental result of [4] is that there is an isomorphism of monoidal W^{∗}-categories

π : R_{0} → TL^{2}(δ) such that π(U^{⊠k}) = k,
π(Mor(U^{⊠k}, U^{⊠l})) = TL_{2k,2l}(δ) (k, l ∈ ℕ).

This is defined on the generating morphisms ν, m, id_{B} of R_{0} by

π(ν) = t(0, 0) = δ^{−1/2} ⋃ ∈ TL_{0,2}(δ),

π(m) = δt(1, 1)^{∗} = δ^{1/2} | ⋂ | ∈ TL_{4,2}(δ), π(id_{B}) = 1_{2} = | | ∈ TL_{2}(δ).

From now on we will omit π in our notation and simply view id_{B}, ν and m as generators of (a concrete faithful representation of) TL^{2}(δ). Using the above isomorphism, a description of the irreducible unitary representations of 𝔾 and their fusion rules can be obtained. See [4] and Theorem 4.1 of [3] for the tracial case.

Theorem 3.8. There exists a bijection Irr(𝔾) ≅ ℕ and a maximal family {U^{k}}_{k∈ℕ} of pairwise inequivalent irreducible finite dimensional unitary representations of 𝔾 such that:

1. \overline{U^{k}} ≅ U^{k} for each k ∈ ℕ.

2. U^{0} = 1_{C_{u}(𝔾)} and the fundamental representation U decomposes as U ≅ U^{0} ⊕ U^{1}.

3. U^{n} ⊠ U^{k} ≅ ⨁_{r=0}^{2 min{n,k}} U^{k+n−r} for all n, k ∈ ℕ. In particular, 𝔾 has the same fusion rules as SO(3).

4. The sequence of representation dimensions {d_{k} = dim(U^{k})}_{k∈ℕ} is given by the recursion

d_{0} = 1,  d_{1} = dim B − 1,  d_{1}d_{k} = d_{k+1} + d_{k} + d_{k−1}  (k ≥ 1).

3.2 Explicit Models

To facilitate our investigation of certain structural properties 𝔾, it will be useful to have explicit models for the irreducible representations {U^{k}}_{k∈ℕ} of 𝔾, as well as concrete expressions for certain morphisms between their tensor products. Consider the kth tensor power U^{⊠k} of the fundamental representation of 𝔾. Using the isomorphism R_{0} = TL^{2}(δ), we can consider the Jones-Wenzl projection p_{2k} ∈ Mor(U^{⊠k}, U^{⊠k}) = TL_{2k}(δ). Put H_{k} = p_{2k}B^{⊗k} and define U^{k} to be the subrepresentation of U^{⊠k} obtained by restricting to the invariant subspace H_{k} ⊂ B^{⊗k}. One can inductively show that {U^{k}}_{k∈ℕ} forms a complete family of pairwise inequivalent irreducible unitary representations of 𝔾 satisfying the hypotheses of Theorem 3.8. For a good introduction to the basic properties of the Jones-Wenzl projections {p_{y}}_{y∈ℕ}, we refer the reader to the book [27].

Using the notation from Section 3.1, note that p_{y} ∈ TL_{y}(δ) is given explicitly by p_{0} = 1, p_{1} = 1_{1}, p_{2} = 1_{2} − νν^{∗} = 1_{2} − t(0, 0)t(0, 0)^{∗}, and (more generally)

p_{y} = 1_{y} − ⋁_{0≤r≤y−2} t(r, y − r − 2)t(r, y − r − 2)^{∗}

=\underbrace{⋯}_{y} − ⋁_{0≤r≤y−2} (δ^{−1}\underbrace{⋯}_{r} \underbrace{⋯}_{y−r−2} ) (y ≥ 2).

Here, ∨ denotes the supremum of a family of projections. Note that a useful consequence of the above formulae for {p_{y}}_{y∈ℕ} is the following absorption property: (p_{x} ⊗ p_{y})p_{x+y} = p_{x+y} ∀x, y ∈ ℕ. This property will be used throughout the paper. Given n, k, l ∈ ℕ such that U^{l} ⊂ U^{n} ⊠ U^{k} (i.e., U^{l} is equivalent to a subrepresentation of U^{n} ⊠ U^{k}), we now proceed to construct explicit morphisms ρ_{l}^{n⊠k} ∈ Mor(U^{l}, U^{n} ⊠ U^{k}) = (p_{2n} ⊗ p_{2k})TL_{2l,2(n+k)}(δ)p_{2l}. To do this we first require some notation.

Notation 3.9. 1. Let δ ≥ 2 and let 0 < q ≤ 1 be such that δ = q + q^{−1}. Recall that the q-numbers and q-factorials are defined for each a ∈ ℕ by

[a]_{q} = \frac{q^{a} − q^{−a}}{q − q^{−1}} = \frac{q^{−a+1}(1 − q^{2a})}{1 − q^{2}}, [a]_{q}! = [a]_{q}[a − 1]_{q} … [1]_{q}.

Of course when q = 1, the above formulas reduce to [a]_{q} = a and [a]_{q}! = a!. Observe that δ = [2]_{q}, [3]_{q} = δ^{2} −1, and [2k +1]_{q} = m_{k} - the quantum dimension of U^{k}.

2. Given a ∈ ℕ, 1_{a} will denote the unit of the algebra TL_{a}(δ). Thus 1_{2a} = id_{B^{⊗a}} via the isomorphism R_{0} ≅ TL^{2}(δ). More generally, if x ≤ a, 1_{x} will denote any collection of x parallel through-strings in a Temperley-Lieb diagram belonging to TL_{a}(δ).

3. For r ≥ 1, let t_{r} = [r + 1]_{q}^{−1/2}(p_{r} ⊗ p_{r})∪^{(r)}, where ∪^{(r)} denotes r nested cups in TL_{0,2r}(δ). In terms of planar diagrams,

t_{r} = [r + 1]_{q}^{−1/2}  \begin{matrix}p_{r} \\ p_{r} \end{matrix}  ∈ TL_{0,2r}(δ).

Now if U^{l} ⊂ U^{n} ⊠ U^{k}, then by Theorem 3.8 there is a unique 0 ≤ r ≤ 2 min{n, k} such that l = n + k − r. Define ρ_{l}^{n⊠k} ∈ Mor(U^{l}, U^{n} ⊠ U^{k}) by setting

ρ_{l}^{n⊠k} = (p_{2n} ⊗ p_{2k})(1_{2n−r} ⊗ t_{r} ⊗ 1_{2k−r})p_{2l}.  (3.2)

Using the results of [27], Section 9.10, it follows that (ρ_{l}^{n⊠k})^{∗}ρ_{l}^{n⊠k} = C_{(n,k,l)}p_{2l}, where

C_{(n,k,l)} = \frac{[2n + 2k − r + 1]_{q}![2n − r]_{q}![r]_{q}![2k − r]_{q}!}{[r + 1]_{q}[2l + 1]_{q}[2n]_{q}![2k]_{q}![2l]_{q}!} > 0.  (3.3)

Therefore C_{(n,k,l)}^{−1/2}ρ_{l}^{n⊠k} is an isometric morphism from H_{l} onto the subspace of H_{n} ⊗ H_{k} equivalent to H_{l}. Moreover, this isometry is unique (up to scalar multiplication by 𝕋) since the inclusion U^{l} ⊂ U^{n} ⊠ U^{k} is multiplicity-free by Theorem 3.8. When we study the property of rapid decay and factoriality, it will be useful for us to have a description of the quantity 1_{2n−r} ⊗ t_{r} ⊗ 1_{2k−r} in formula (3.2) in terms of the basic morphisms m, ν, 1_{2} and the Jones-Wenzl projections {p_{y}}_{y∈ℕ}. This can be done recursively as follows: if r = 2s is even, then t_{2s} is given by the initial condition and recursion

t_{0} = 1 ∈ ℂ,  t_{2} = [3]_{q}^{−1/2}(p_{2} ⊗ p_{2})(m^{∗}ν),  p_{2} = 1_{2} − νν^{∗},  (3.4)
t_{2(k+l)} = (\frac{[2k + 1]_{q}[2l + 1]_{q}}{[2k + 2l + 1]_{q}} )^{1/2}(p_{2k+2l} ⊗ p_{2l+2k})(1_{2k} ⊗ t_{2l} ⊗ 1_{2k})t_{2k}. (3.5)

If r = 2s + 1 is odd, then

1_{2n−r} ⊗ t_{r} ⊗ 1_{2k−r} = (\frac{[2s + 1]_{q}}{[2s + 2]_{q}[2]_{q}} )^{1/2}(1_{2n−2s−1} ⊗ p_{2s+1} ⊗ p_{2s+1} ⊗ 1_{2k−2s−1}) (3.6)

× (1_{2n−2s−2} ⊗ (1_{2} ⊗ t_{2s} ⊗ 1_{2})m^{∗} ⊗ 1_{2k−2s−2}),

which can be readily verified using the fact that m^{∗} = [2]_{q}(1_{1} ⊗ t_{1} ⊗ 1_{1}). Note furthermore that C_{(k,k,0)} = 1 for each k ∈ ℕ by equation (3.3). Thus each t_{2k} ∈ Mor(1, U^{k} ⊠ U^{k}) is an isometric morphism, and ρ_{l}^{n⊠k} is a contraction (being a composition of contractions). To conclude this section let us also fix once and for all an orthonormal basis {e_{i}}_{i=1}^{d_{1}} for H_{1} = B ⊖ ℂ1 and write U^{1} = [u_{ij}^{1}] relative to this basis. The equivalence U^{1} ≅ \overline{U^{1}} given by Theorem 3.8 ensures that there is an invertible matrix F_{1} ∈ GL(d_{1}, ℂ) such that U^{1} = (F_{1} ⊗ 1)\overline{U^{1}}(F_{1}^{−1} ⊗ 1). Moreover, since U^{1} is irreducible, we can assume that \overline{F_{1}}F_{1} = c1 where c = ±1 (see [1], Page 242). By possibly replacing F_{1} by zF_{1} for some z ∈ 𝕋, we may also assume that

t_{2} = Tr(F_{1}^{∗}F_{1})^{−1/2} ∑_{i=1}^{d_{1}} e_{i} ⊗ F_{1}e_{i} ∈ Mor(1, U^{1} ⊠ U^{1}).  (3.7)

This follows because both sides of equation (3.7) are evidently isometries in the one dimensional space Mor(1, U^{1} ⊠ U^{1}). Finally, note that when ψ is the δ-trace on B, then 𝔾_{aut}(B, ψ) is of Kac-type. We therefore can and will assume that F_{1} ∈ U(d_{1}) is unitary. Also note that when ψ is the δ-trace, formula (3.4) yields σt_{2} = t_{2}, where σ is the tensor flip-map. On the other hand, formula (3.7) yields σt_{2} = ct_{2}. Consequently \overline{F_{1}}F_{1} = 1. We leave the easy details to the reader.

4 Approximation Properties and The Property of Rapid Decay

This section is devoted to proving some approximation properties for the reduced operator algebras and convolution algebras associated to the quantum groups 𝔾_{aut}(B, ψ), where ψ is the δ-trace on B.

4.1 The Haagerup Property

Our first goal is to prove that L^{∞}(𝔾) has the Haagerup property. We begin by recalling this approximation property.

Definition 4.1. A finite von Neumann algebra (M, τ) has the Haagerup property if there exists a net {Φ_{t}}_{t∈Λ} of normal unital completely positive τ-preserving maps on M such that

1. For each t ∈ Λ, the L^{2}-extension Φ̂_{t} : L^{2}(M) → L^{2}(M) is a compact operator.

2. For each ξ ∈ L^{2}(M), lim_{t∈Λ} ∥Φ̂_{t}ξ − ξ∥_{L^{2}(M)} = 0.

Note that the Haagerup property does not actually depend on the choice of faithful normal trace τ : M → ℂ [26].

Theorem 4.2. Let B be a finite dimensional C^{∗}-algebra with δ-trace ψ and let 𝔾 = 𝔾_{aut}(B, ψ). Then L^{∞}(𝔾) has the Haagerup property.

To prove Theorem 4.2, we begin by recalling a standard method for constructing normal completely positive maps on L^{∞}(𝔾) from states on C_{u}(𝔾). See for example [14], Lemma 3.4.

Proposition 4.3. Let 𝔾 be a compact quantum group and φ ∈ C_{u}(𝔾)^{∗} be a state. Then there exists a unique normal unital completely positive h-preserving map M_{φ} ∈ CB(L^{∞}(𝔾)) defined by

M_{φ}λ(a) = λ((φ ⊗ id_{C_{u}(𝔾)})Δa) (a ∈ C_{u}(𝔾)).

In particular, M_{φ}C_{r}(𝔾) ⊆ C_{r}(𝔾).

Remark 4.4. It is easily verified that the map M_{φ} defined in Proposition 4.3 satisfies the relation Δ_{r}M_{φ} = (M_{φ} ⊗ id_{L^{∞}(𝔾)})Δ_{r}. A map T ∈ CB(L^{∞}(𝔾)) such that Δ_{r}T = (T ⊗ id_{L^{∞}(𝔾)})Δ_{r} (respectively Δ_{r}T = (id_{L^{∞}(𝔾)} ⊗ T)Δ_{r}) is called a completely bounded left (respectively right) multiplier of L^{∞}(𝔾). This terminology originates from the classical situation where L^{∞}(𝔾) = V N(Γ) is the reduced von Neumann algebra of a discrete group Γ. In this case, the above definition reduces to the one for completely bounded Fourier multipliers on Γ. See [16]. Note that for (unital) completely positive left multipliers, a recent result of Daws [17] shows that Proposition 4.3 is actually a characterization of such normal multipliers.

Now let {U^{α} = [u_{ij}^{α}] : α ∈ Irr(𝔾)} be a maximal family of irreducible unitary representations of a compact quantum group 𝔾 and let χ_{α} = (Tr⊗id_{C_{u}(𝔾)})U^{α} ∈ C_{u}(𝔾) be the character of the representation U^{α}. We say that T ∈ CB(L^{∞}(𝔾)) is a central completely bounded multiplier if T is both a left and right completely bounded multiplier of L^{∞}(𝔾). This is equivalent to saying that for each α ∈ Irr(𝔾), there is a constant c_{α} ∈ ℂ such that T λ(u_{ij}^{α}) = c_{α}λ(u_{ij}^{α}) for all 1 ≤ i, j ≤ d_{α}. The following result (proved in [14]) will be our main tool for proving Theorem 4.2. It shows that when 𝔾 is of Kac type, central completely positive multipliers can be obtained from states on the (generally much smaller and more tractable) subalgebra C^{∗}(χ_{α} : α ∈ Irr(𝔾)) ⊂ C_{u}(𝔾) generated by the irreducible characters of 𝔾.

Theorem 4.5 ([14], Theorem 3.7). Let 𝔾 be a compact quantum group of Kac type and let φ ∈ C^{∗}(χ_{α} : α ∈ Irr(𝔾))^{∗} be a state. Then there exists a unique normal unital completely positive h-preserving map T_{φ} ∈ CB(L^{∞}(𝔾)) defined by

T_{φ}λ(u_{ij}^{α}) = \frac{φ(χ_{α}^{∗})}{d_{α}}λ(u_{ij}^{α}) (α ∈ Irr(𝔾), 1 ≤ i, j ≤ d_{α}).

Moreover, T_{φ}C_{r}(𝔾) ⊆ C_{r}(𝔾).

Remark 4.6. Our proof in [14] of Theorem 4.5 uses the traciality of the Haar state in an essential way. Indeed, the proof relies on both the boundedness of the coinverse κ : A → A and the existence of an h-preserving conditional expectation E : L^{∞}(𝔾)\overline{⊗}L^{∞}(𝔾) → Δ_{r}(L^{∞}(𝔾)). Each of these conditions imply that h is tracial.

Let us now return to the quantum automorphism groups 𝔾 = 𝔾_{aut}(B, ψ). Theorem 4.5 suggests that we study the (state space of the) C^{∗}-algebra C^{∗}(χ_{k} : k ∈ ℕ) ⊂ C_{u}(𝔾), where χ_{k} is the character of the irreducible representation U^{k} given by Theorem 3.8. Since 𝔾 has commutative fusion rules and each irreducible representation U^{k} is equivalent to \overline{U^{k}} by Theorem 3.8, it follows from general character theory that C^{∗}(χ_{k} : k ∈ ℕ) is commutative and that each character χ_{k} is self-adjoint. In the following paragraphs we will obtain a more precise description of this C^{∗}-algebra. Let {S_{k}}_{k=0}^{∞} denote the (dilated) Chebyshev polynomials of the second kind, which are defined by the recursion

S_{0}(x) = 1,  S_{1}(x) = x,  S_{1}(x)S_{k}(x) = S_{k+1}(x) + S_{k−1}(x)  (k ≥ 1).

These are the monic orthogonal polynomials for Wigner’s semicircle law, which is the probability measure supported on [−2, 2] with density \frac{\sqrt{4−x^{2}}}{2π}. Following Section 4 of [29], we let {Π_{k}}_{k=0}^{∞} be the family of polynomials defined by Π_{0}(x) = 1 and Π_{k}(x) = S_{k}(x − 2) + S_{k−1}(x − 2) for each k ≥ 1. Then we have the recursion

Π_{1}(x)Π_{k}(x) = Π_{k+1}(x) + Π_{k}(x) + Π_{k−1}(x)  (k ≥ 1).  (4.1)

The polynomials {Π_{k}}_{k∈ℕ} are the monic orthogonal polynomials for the free Poisson law (with parameter 1), which is the probability measure supported on [0, 4] with density given by \frac{\sqrt{4x−x^{2}}}{2πx}. Observe that one can equivalently define the family \{Π_{k}\}_{k∈ℕ} by setting Π_{k}(x) = S_{2k}(\sqrt{x}). To see this, note that S_{2k}(x) is a polynomial of degree 2k which contains only terms of even degree. Therefore we can define a degree k polynomial Π̃_{k} by setting Π̃_{k}(x) = S_{2k}(\sqrt{x}). It is then easy to verify that Π̃_{0} = Π_{0} = 1, Π̃_{1}(x) = x − 1 = Π_{1}(x), and that the Π̃_{k}’s satisfy the recursion (4.1).

Proposition 4.7. Let χ ∈ C_{u}(𝔾) be the character of the fundamental representation U of 𝔾 = 𝔾_{aut}(B, ψ). Then χ = χ^{∗} and there is a ∗-isomorphism

Ψ : C^{∗}(χ_{k} : k ∈ ℕ) → C(σ)  given by  Ψ : χ_{k} ↦ Π_{k}|_{σ},

where σ := σ(χ) ⊂ ℝ is the spectrum of χ.

Proof. By Theorem 3.8, U ≅ 1 ⊕ U^{1} and each χ_{k} is self-adjoint. Therefore χ = 1 + χ_{1} = (1 + χ_{1})^{∗} = χ^{∗}. Moreover, the fusion rules for {U^{k}}_{k∈ℕ} yield the character relations χ_{1}χ_{k} = χ_{k+1} + χ_{k} + χ_{k−1} for all k ≥ 1. Thus C^{∗}(χ_{k} : k ∈ ℕ) = C^{∗}(1, χ). Let Ψ : C^{∗}(χ_{k} : k ∈ ℕ) = C^{∗}(1, χ) → C(σ) be the Gelfand isomorphism. Then for any t ∈ σ and k ≥ 1, we have Ψχ_{1}(t)Ψχ_{k}(t) = Ψ(χ_{1}χ_{k})(t) = Ψχ_{k+1}(t) + Ψχ_{k}(t) + Ψχ_{k−1}(t), with the initial conditions Ψχ_{0}(t) = Ψ(1_{C_{u}(𝔾)})(t) = 1 and Ψχ_{1}(t) = Ψ(χ − χ_{0})(t) = t − 1. Comparing with the initial conditions and recursion (4.1) for the polynomials {Π_{k}}_{k∈ℕ}, we obtain Ψχ_{k} = Π_{k}|_{σ} (k ∈ ℕ).

This next proposition gives us some useful information about the spectrum σ ⊂ ℝ of χ ∈ C_{u}(𝔾), at least in the δ-trace case.

Proposition 4.8. Let ψ be the δ-trace on B and let σ = σ(χ) ⊂ ℝ be as in Proposition 4.7. Then [0, dim B] ⊆ σ.

Remark 4.9. We suspect that the equality [0, dim B] = σ always holds in Proposition 4.8, but we are unable to show that χ ≥ 0 in general.

Proof. Let μ be the spectral measure of χ relative to the Haar state h. Since the irreducible characters of a compact quantum group form an orthonormal family with respect to the L^{2}(𝔾)-inner product, Proposition 4.7 yields

∫_{supp(μ)} Π_{k}(t)Π_{l}(t)dμ(t) = h(Π_{k}(χ)^{∗}Π_{l}(χ)) = h(χ_{k}^{∗}χ_{l}) = δ_{kl} (k, l ∈ ℕ).

I.e., {Π_{k}}_{k∈ℕ} are the monic orthogonal polynomials for μ. Therefore μ must be the free Poisson law, and in particular [0, 4] = supp(μ) ⊆ σ.

We now show that [4, dim B] ⊂ σ. Write B as a direct sum of matrix algebras, say

B = C(X_{0}) ⨁ ⨁_{1≤i≤k, n_{i}≥2} M_{n_{i}}(ℂ),

where C(X_{0}) denotes the maximal commutative ideal of B (which may be zero). In what follows, we will make extensive use of the fact that 𝔾 contains (ℤ̂_{2}^{∗⌊|X_{0}|/2⌋}) × Π_{1≤i≤k}SO(n_{i}) as a quantum subgroup. Let n_{0} = |X_{0}| so that dim B = n_{0} + ∑_{i=1}^{k} n_{i}^{2}. We will assume for the remainder of the proof that both of the components C(X_{0}) and ⨁_{1≤i≤k, n_{i}≥2} M_{n_{i}}(ℂ) are non-zero. (The trivial modification of our argument for the remaining two cases is left to the reader). Let 𝔾_{0} = S_{n_{0}}^{+} and let 𝔾_{i} = SO(n_{i}) for 1 ≤ i ≤ k. For each i, let β_{i} : B_{i} → B_{i} ⊗C_{u}(𝔾_{i}) be the canonical action of 𝔾_{i} on (B_{i}, ψ_{i}), where B_{i} is the ith direct summand of B and ψ_{i} = ψ|_{B_{i}}. (In other words, β_{0} is the fundamental action of S_{n_{0}}^{+} on C(X_{0}) and β_{i} is the standard action of SO(n_{i}) on M_{n_{i}}(ℂ) by inner automorphisms.) Now let 𝔾′ = ⨂_{i=0}^{k} 𝔾_{i} be the tensor product quantum group (see [35], Chapter 6) and consider the diagonal action

β : B → B ⊗ C_{u}(𝔾′); β = ⨁_{0≤i≤k}(id_{B} ⊗ ι_{i})β_{i},

of 𝔾′ on (B, ψ), where ι_{i} : C_{u}(𝔾_{i}) ↪ ℂ_{u}(𝔾′) is the canonical embedding. Invoking Definition 3.2, we see that there exists a unital ∗-homomorphism π : C_{u}(𝔾) → C_{u}(𝔾′) such that β = (id_{B} ⊗ π)α, where α : B → B ⊗ C_{u}(𝔾) is the universal action of 𝔾 on (B, ψ). Moreover, since β is faithful in the sense of [40], it follows that π is surjective. In particular, if U′ denotes the representation of 𝔾′ associated to the action β and χ′ is its character, then χ′ = π(χ) and σ(χ′) ⊆ σ. It therefore suffices to show that [4, dim B] ⊆ σ(χ′). Let χ_{(0)} ∈ C_{u}(𝔾_{0}) denote the fundamental character of 𝔾_{0}, and let χ_{(i)} ∈ C(SO(n_{i})) denote the character of the standard representation of SO(n_{i}) on ℂ^{n_{i}}. A simple calculation using the definition of 𝔾′ and the action β shows that

χ′ = χ_{(0)} ⊗ \underbrace{1 ⊗ … ⊗ 1}_{k times} + ∑_{i=1}^{k} \underbrace{1 ⊗ … ⊗ 1}_{i times} ⊗χ_{(i)}^{2} ⊗ \underbrace{1 ⊗ … ⊗ 1}_{k−i times},

and therefore (by tensor independence) σ(χ′) = σ(χ_{(0)}) + ∑_{i=1}^{k} σ(χ_{(i)}^{2}). Moreover, an elementary exercise in linear algebra shows that for i ≥ 1, χ_{(i)} = χ_{(i)}^{∗}, ∥χ_{(i)}∥ = n_{i} and [0, n_{i}] ⊂ σ(χ_{(i)}). Therefore

σ(χ′) = σ(χ_{(0)}) + ∑_{i=1}^{k}[0, n_{i}^{2}] = σ(χ_{(0)}) + [0, dim B − n_{0}].  (4.2)

To finish the proof, we analyze the spectrum σ(χ_{(0)}) of χ_{(0)} ∈ C_{u}(𝔾_{0}). If 1 ≤ n_{0} ≤ 4, then 4 ≥ n_{0} = ϵ(χ_{(0)}) ∈ σ(χ_{0}), and (4.2) yields [4, dim B] ⊂ [n_{0}, dim B] ⊆ σ(χ′). If n_{0} ≥ 5, write n_{0} = 2s + r where s ≥ 2 and r ∈ {0, 1}. Since C_{u}(𝔾_{0}) = C_{u}(S_{n_{0}}^{+}) is the universal C^{∗}-algebra generated by an n_{0} × n_{0} magic unitary matrix [v_{ij}], there exists a unique surjective ∗-homomorphism π_{0} : C_{u}(𝔾_{0}) → ∗_{l=1}^{s}C^{∗}(ℤ_{2}) which sends [v_{ij}] to the block-diagonal matrix W ∈ M_{n_{0}}( ∗_{l=1}^{s} C^{∗}(ℤ_{2})) given by

W = \{ \begin{matrix}⨁_{j=1}^{s} ( \begin{matrix}p_{j} & 1 − p_{j} \\ 1 − p_{j} & p_{j} \end{matrix} ) & if r = 0, \\ 1 ⊕ ⨁_{j=1}^{s} ( \begin{matrix}p_{j} & 1 − p_{j} \\ 1 − p_{j} & p_{j} \end{matrix} ) & if r = 1. \end{matrix}

Here p_{j} is the projection \frac{1}{2}(1 − g_{j}), where g_{j} is the canonical unitary generator of the jth copy of C^{∗}(ℤ_{2}) in the free product ∗_{l=1}^{s}C^{∗}(ℤ_{2}). We then have π_{0}(χ_{(0)}) = r1 + ∑_{j=1}^{s} 2p_{j}. Since the spectrum of a sum of s free projections is well known to equal [0, s] (see [18], Example 4), we obtain σ(π_{0}(χ_{(0)})) = [r, 2s+r] ⊂ σ(χ_{(0)}). Using this last inclusion in (4.2), we conclude that σ(χ′) ⊃ [r, dim B] ⊃ [4, dim B].

We are now ready to prove Theorem 4.2.

Proof of Theorem 4.2. If 1 ≤ dim B ≤ 4 then 𝔾 is co-amenable by [3], Corollary 4.2, and it follows from [11], Theorem 1.1, that L^{∞}(𝔾) is injective. In particular, L^{∞}(𝔾) has the Haagerup property. Assume now that dim B ≥ 5 and fix 4 < t_{0} < 5. From the isomorphism C^{∗}(χ_{k} : k ∈ ℕ) ≅ C(σ) given by Proposition 4.7 together with the fact that [0, dim B] ⊆ σ from Proposition 4.8, it follows that for each t ∈ [t_{0}, dim B), there is a unique state φ_{t} ∈ C^{∗}(χ_{k} : k ∈ ℕ)^{∗} given by

φ_{t}(χ_{k}) = Π_{k}(t) (k ∈ ℕ).

Consider the irreducible representation U^{k} = [u_{ij}^{k}] of 𝔾 with label k. From part (4) of Theorem 3.8 and the recursion (4.1), we have d_{k} = Π_{k}(dim B) for all k ∈ ℕ. Applying Theorem 4.5 to the family of states {φ_{t}}_{t∈[t_{0},dim B)} ⊂ C^{∗}(χ_{k} : k ∈ ℕ)^{∗}, we obtain a net of central completely positive multipliers {Φ_{t}}_{t∈[t_{0},dim B)} ⊂ CB(L^{∞}(𝔾)) defined by

Φ_{t}(λ(u_{ij}^{k})) = \frac{Π_{k}(t)}{Π_{k}(dim B)}λ(u_{ij}^{k})  (k ∈ ℕ, 1 ≤ i, j ≤ d_{k}).  (4.3)

Here [t_{0}, dim B) ⊂ ℝ is directed in the usual way. We now show that this net of normal unital completely positive maps is L^{2}- compact and converges pointwise in L^{2}-norm to the identity map. Using the decomposition L^{2}(𝔾) = ⨁_{k∈ℕ} L_{k}^{2}(𝔾) given by (2.3), each L^{2}-extension Φ̂_{t} ∈ B(L^{2}(𝔾)) is given by Φ̂_{t} = ∑_{k∈ℕ} \frac{Π_{k}(t)}{Π_{k}(dim B)}P_{k}. Since lim_{t→dim B} \frac{Π_{k}(t)}{Π_{k}(dim B)} = 1

for each k and ∥Φ̂_{t}∥ = 1 for each t, it follows that lim_{t→dim B} ∥Φ̂_{t}ξ −ξ∥_{L^{2}(𝔾)} = 0 for each ξ ∈ L^{2}(𝔾). To see that each Φ̂_{t} is compact, observe that since Φ̂_{t} is the orthogonal direct sum of the finite rank projections {P_{k}}_{k∈ℕ}, it suffices to show that \{\frac{Π_{k}(t)}{Π_{k}(dim B)}\}_{k∈ℕ} ∈ c_{0}(ℕ) for each t ∈ [t_{0}, dim B). To check this last fact,

note that 0 < \frac{Π_{k}(t)}{Π_{k}(dim B)} = \frac{S_{2k}(\sqrt{t})}{S_{2k}(\sqrt{dim B})}, and an easy exercise (see for example [14], Proposition 4.4) shows that there is a constant A(t_{0}) > 0 (depending only on t_{0}) such that

\frac{S_{2k}(\sqrt{t})}{S_{2k}(\sqrt{dim B})} ≤ A(t_{0})(\begin{matrix}√ & \overline{t} \\ √ & \overline{dim B} \end{matrix} )^{2k} = A(t_{0})(\frac{t}{dim B} )^{k} ⟶ 0 (k → ∞).

(4.4)

4.2 The Property of Rapid Decay

Given a finitely generated discrete group Γ with symmetric generating set S = {γ_{1}, …, γ_{s}}, there is a natural length function ℓ = ℓ_{S} : Γ → ℕ defined by assigning to each γ ∈ Γ the length of its shortest representation as a reduced word in the elements of S. The group Γ is said to have the property of rapid decay (with respect to ℓ), or just property RD, if there exists a polynomial P ∈ ℝ_{+}[x] such that for each n ∈ ℕ and each function f : {γ ∈ Γ | ℓ(γ) = n} → ℂ supported on words in Γ of length n, the following inequality holds

∥λ(f)∥_{V N(Γ)} ≤ P (n)∥f∥_{ℓ^{2}(Γ)}.

It is a standard fact that property RD is essentially independent of the initial generating set S for Γ. Perhaps the most famous example of a group with property RD is the non-abelian free group 𝔽_{d} on d ≥ 2 generators, which was shown by Haagerup [24] to have property RD with P (x) = x + 1. The notion of property RD for discrete quantum groups was defined and studied by Vergnioux in [38]. There he showed that property RD can only occur for unimodular discrete quantum groups, and he also exhibited the first (and what currently appear to be the only) truly “quantum” examples having this property. The goal in this section is to prove that if B is a finite dimensional C^{∗}-algebra with δ-trace ψ, then the dual unimodular discrete quantum group 𝔾̂ of 𝔾 = 𝔾_{aut}(B, ψ) has property RD. We will use this result to obtain some finite rank approximation properties for L^{∞}(𝔾), C_{r}(𝔾) and L^{1}(𝔾) in Theorem 4.15. Property RD will also be a key tool in our analysis of simplicity and uniqueness of trace for C_{r}(𝔾) in Section 5.3. For our purposes, property RD for 𝔾̂ will be most conveniently formulated in dual terms, as an equivalent property of L^{∞}(𝔾). More precisely, we will prove the following theorem.

Theorem 4.10. Let ψ be the δ-trace on a finite dimensional C^{∗}-algebra B and let C_{n}(𝔾) = span{(ω_{η,ξ} ⊗ id)U^{n} : ξ, η ∈ H_{n}} ⊂ C_{u}(𝔾) be the subspace of matrix elements of the nth irreducible representation of 𝔾. Then there exists a constant D > 0 (depending only on dim B) such that

∥λ(a)∥_{L^{∞}(𝔾)} ≤ D(2n + 1)∥Λ_{h}(a)∥_{L^{2}(𝔾)} (n ∈ ℕ, a ∈ C_{n}(𝔾)).

In particular, each orthogonal projection P_{n} : L^{2}(𝔾) → L_{n}^{2}(𝔾) is bounded as a map from L^{2}(𝔾) into C_{r}(𝔾) with ∥P_{n}∥_{L^{2}→L^{∞}} ≤ D(2n + 1).

Remark 4.11. Note that the O(2n + 1) growth rate of the constants in Theorem 4.10 is optimal. To see this, consider the character χ_{n} ∈ C_{n}(𝔾) of the irreducible representation U^{n}. Using the notation and arguments from the proof of Proposition 4.8, it follows that ∥λ(χ_{n})∥ = sup_{t∈[0,4]} |Π_{n}(t)| = sup_{t∈[0,4]} |S_{2n}(\sqrt{t})| = S_{2n}(2) = 2n + 1. Noting that ∥Λ_{h}(χ_{n})∥_{L^{2}(𝔾)} = 1, we see that the O(2n + 1) bound is actually obtained by the sequence {λ(χ_{n})}_{n∈ℕ}.

Our proof of Theorem 4.10 is based on the ideas in Section 4 of [38], where property RD is proved for the duals of free orthogonal and free unitary quantum groups. The main difference between the present situation and the one in [38] is that for quantum automorphism groups, the fusion rules U^{n} ⊠ U^{k} ≅ ⨁_{r=0}^{2 min{n,k}} U^{k+n−r} imply that each tensor product U^{n} ⊠ U^{k} contains both even and odd length subrepresentations, whereas in the orthogonal case all subrepresentations of U^{n} ⊠ U^{k} have the same parity. This has the effect of complicating some norm estimates.

To begin the proof of Theorem 4.10, note that it suffices to show that there is a constant D > 0 (depending only on dim B) such that for each n ∈ ℕ and each a ∈ C_{n}(𝔾),

∥P_{l}λ(a)P_{k}∥_{B(L^{2}(𝔾))} ≤ D∥Λ_{h}(a)∥_{L^{2}(𝔾)}  (k, l ∈ ℕ).  (4.5)

To prove this, we follow along the lines of Haagerup [24], where an analogous reduction was obtained for free groups. Assume that (4.5) holds and take n ∈ ℕ, a ∈ C_{n}(𝔾) and ξ ∈ L^{2}(𝔾). Using the fusion rules from Theorem 3.8, it is not difficult to see that the sets A_{l,n} = {k ∈ ℕ : U^{l} ⊂ U^{n} ⊠ U^{k}} and B_{k,n} = {l ∈ ℕ : U^{l} ⊂ U^{n}⊠U^{k}} both have cardinality at most 2n+1. Therefore

∥λ(a)ξ∥_{2}^{2} = ∑_{l∈ℕ} ∥P_{l}λ(a)ξ∥_{2}^{2} = ∑_{l∈ℕ} ∥ ∑_{k∈ℕ: U^{l}⊂U^{n}⊠U^{k}} P_{l}λ(a)P_{k}ξ∥_{2}^{2}

≤ ∑_{l∈ℕ} ( ∑_{k∈ℕ: U^{l}⊂U^{n}⊠U^{k}} ∥P_{k}λ(a)P_{k}ξ∥)^{2}

≤ ∑_{l∈ℕ} ( ∑_{k∈ℕ: U^{l}⊂U^{n}⊠U^{k}} D∥Λ_{h}(a)∥_{2}∥P_{k}ξ∥_{2})^{2}

≤ D^{2}∥Λ_{h}(a)∥_{2}^{2} ∑_{l∈ℕ} |A_{l,n}|( ∑_{k∈ℕ: U^{l}⊂U^{n}⊠U^{k}} ∥P_{k}ξ∥_{2}^{2})

≤ D^{2}∥Λ_{h}(a)∥_{2}^{2}(2n + 1) ∑_{k∈ℕ} ( ∑_{l∈ℕ: U^{l}⊂U^{n}⊠U^{k}} ∥P_{k}ξ∥_{2}^{2})

= D^{2}∥Λ_{h}(a)∥_{2}^{2}(2n + 1) ∑_{k∈ℕ} |B_{k,n}|∥P_{k}ξ∥_{2}^{2} ≤ D^{2}∥Λ_{h}(a)∥_{2}^{2}(2n + 1)^{2}∥ξ∥_{2}^{2},

which yields ∥λ(a)∥_{L^{∞}(𝔾)} ≤ D(2n + 1)∥Λ_{h}(a)∥_{L^{2}(𝔾)}. The remainder of this section is devoted to proving (4.5). To do this, we utilize the Fourier transform defined by Podleś and Woronowicz in Section 2 of [32]. Let 𝔾 be a compact quantum group and let 𝕍 ∈ M(c_{0}(𝔾̂) ⊗ C_{u}(𝔾)) be the unitary given by (2.5). Define

F : C_{alg}(𝔾) → ℂ[𝔾̂];  F(a) = (id ⊗ ah)𝕍^{∗}  (a ∈ C_{u}(𝔾)),  (4.6)

where ah ∈ C_{u}(𝔾)^{∗} is given by ah(b) := h(ba), (b ∈ C_{u}(𝔾)). Using the descriptions of h and ĥ_{L} given by (2.2) and (2.6), it is readily checked that F : C_{alg}(𝔾) → ℂ[𝔾̂] is a linear bijection with inverse

F^{−1} : ℂ[𝔾̂] → C_{alg}(𝔾);  F^{−1}(x) = (xĥ_{L} ⊗ id)𝕍  (x ∈ ℂ[𝔾̂]), (4.7)

where xĥ_{L}(y) := ĥ_{L}(yx), (y ∈ ℓ^{∞}(𝔾̂)). Indeed, a simple calculation shows that F(u_{ij}^{α}) = m_{α}^{−1} ∑_{1≤r≤d_{α}}(Q_{α}^{−1})_{ir}e_{jr}^{α}, where {e_{ij}^{α}} denotes the system of matrix units for B(H_{α}) associated to the matrix representation U^{α} = [u_{ij}^{α}]. This calculation also shows that F extends to a unitary isomorphism F : L^{2}(𝔾) → ℓ^{2}(𝔾̂) := L^{2}(ℂ[𝔾̂], ĥ_{L}). Recall that one can define a convolution product Conv : ℂ[𝔾̂] ⊗_{alg} ℂ[𝔾̂] → ℂ[𝔾̂] using the dual coproduct Δ̂ as follows. Given x, y ∈ ℂ[𝔾̂], Conv(x ⊗ y) ∈ ℂ[𝔾̂] is defined to be the unique element, denoted by x ∗ y, such that (x ∗ y)ĥ = (xĥ_{L} ⊗ yĥ_{L}) ∘ Δ̂. With respect to the usual algebra structure on C_{alg}(𝔾) and the convolution product on ℂ[𝔾], the Fourier transform F : C_{alg}(𝔾) → (ℂ[𝔾̂], Conv) becomes an algebra isomorphism. Indeed, since (Δ̂ ⊗ id)𝕍 = 𝕍_{13}𝕍_{23}, we have

F^{−1}(x ∗ y) = (xĥ_{L} ⊗ yĥ_{L} ⊗ id)(Δ̂ ⊗ id)𝕍 = (xĥ_{L} ⊗ yĥ_{L} ⊗ id)(𝕍_{13}𝕍_{23})

= (xĥ_{L} ⊗ id)𝕍 · (yĥ_{L} ⊗ id)𝕍 = F^{−1}xF^{−1}y.

As a final useful observation, we note that FP_{l}F^{−1} = P̂_{l}, where {P̂_{l}}_{l∈ℕ} are the canonical minimal central projections in ℓ^{∞}(𝔾̂) = ∏_{l∈ℕ} B(H_{l}) ⊂ B(ℓ^{2}(𝔾̂)). In view of the above properties of the Fourier transform, we can transfer our required condition (4.5) to an equivalent one on the dual quantum group 𝔾̂. Namely, we must find a D > 0 (depending only on dim B) such that

∥P̂_{l}(x ∗ y)∥_{ℓ^{2}(𝔾̂)} ≤ D∥x∥_{ℓ^{2}(𝔾̂)}∥y∥_{ℓ^{2}(𝔾̂)} (x ∈ P̂_{n}ℂ[𝔾̂], y ∈ P̂_{k}ℂ[𝔾̂], l ∈ ℕ). (4.8)

The following useful result provides an alternate expression for the left-hand side of (4.8) and can be found in [38], Lemma 4.6. For the convenience of the reader we include a proof.

Lemma 4.12. Let 𝔾 = 𝔾_{aut}(B, ψ), where ψ is a δ-form on B, and let w ∈ P̂_{n}ℂ[𝔾̂] ⊗ P̂_{k}ℂ[𝔾̂]. Then for any l ∈ ℕ such that U^{l} ⊂ U^{n} ⊠ U^{k}, we have

∥P̂_{l}Conv(w)∥_{ℓ^{2}(𝔾̂)} = (m_{n}m_{k}m_{l}^{−1})^{1/2}∥Δ̂(P̂_{l})wΔ̂(P̂_{l})∥_{ℓ^{2}(𝔾̂)⊗ℓ^{2}(𝔾̂)}.

Proof. Since the inclusion U^{l} ⊂ U^{n} ⊠ U^{k} is multiplicity-free by Theorem 3.8, there exists a unique z ∈ P̂_{l}ℂ[𝔾̂] such that Δ̂(P̂_{l})wΔ̂(P̂_{l}) = Δ̂(z)(P̂_{n} ⊗ P̂_{k}). (I.e., z = C_{(n,k,l)}^{−1}(ρ_{l}^{n⊠k})^{∗}wρ_{l}^{n⊠k} using the notation of (3.2)–(3.3)). Then from

the definition of convolution, we have P̂_{l}Conv(w) = Conv(Δ̂(P̂_{l})wΔ̂(P̂_{l})) = Conv(Δ̂(z)(P̂_{n} ⊗ P̂_{k})) = z(P̂_{n} ∗ P̂_{k}) = λz for some λ ∈ ℂ. The last equality follows because P̂_{n} ∗ P̂_{k} is evidently central. This yields

∥P̂_{l}Conv(w)∥_{ℓ^{2}(𝔾̂)}^{2}
= ĥ_{L}((P̂_{l}Conv(w))^{∗}P̂_{l}Conv(w)) = λ̄ĥ_{L}(z^{∗}Conv(w))
= λ̄ĥ_{L} ⊗ ĥ_{L}(Δ̂(z)^{∗}w) = λ̄ĥ_{L} ⊗ ĥ_{L}((P̂_{n} ⊗ P̂_{k})Δ̂(z)^{∗}wΔ̂(P̂_{l}))
= λ̄ĥ_{L} ⊗ ĥ_{L}((Δ̂(P̂_{l})wΔ̂(P̂_{l}))^{∗}wΔ̂(P̂_{l})) = λ̄∥Δ̂(P̂_{l})wΔ̂(P̂_{l})∥_{ℓ^{2}(𝔾̂)^{⊗2}}

Finally, the appropriate value of λ is obtained from the equation

λm_{l}^{2}

= λh_{L}(P̂_{l}) = h_{L}(P̂_{l}(P̂_{n} ∗ P̂_{k})) = m_{n}m_{k}(Tr_{n} ⊗ Tr_{k})((Q_{n}P̂_{n} ⊗ Q_{k}P̂_{k})Δ̂(P̂_{l}))

= m_{n}m_{k}(Tr_{n} ⊗ Tr_{k})(Δ̂(Q_{l}P̂_{l})) = m_{n}m_{k}Tr_{l}(Q_{l}) = m_{n}m_{k}m_{l}.

We now restrict to the unimodular case, so that Q_{k} = id_{H_{k}} and m_{k} = d_{k} = dim U^{k} for all k. In this case, we have, ∥x∥_{ℓ^{2}(𝔾̂)} = d_{n}^{1/2}∥x∥_{HS(H_{n})} and

∥Δ̂(P̂_{l})(x⊗y)Δ̂(P̂_{l})∥_{ℓ^{2}(𝔾̂)⊗ℓ^{2}(𝔾̂)} = (d_{n}d_{k})^{1/2}∥Δ̂(P̂_{l})(x⊗y)Δ̂(P̂_{l})∥_{HS(H_{n}⊗H_{k})} for

each x ∈ P̂_{n}ℂ[𝔾̂], y ∈ P̂_{k}ℂ[𝔾̂] (where ∥ · ∥_{HS(·)} denotes the usual Hilbert-Schmidt norm). Combining this fact with Lemma 4.12, we can recast our required inequality (4.8) in terms of Hilbert-Schmidt norms. Thus to prove Theorem 4.10, it suffices to establish the following proposition.

Proposition 4.13. There is a constant D > 0 depending only on dim B such that for every n, k, l ∈ ℕ,

∥Δ̂(P̂_{l})(x ⊗ y)Δ̂(P̂_{l})∥_{HS(H_{n}⊗H_{k})} ≤ D(d_{l}d_{n}^{−1}d_{k}^{−1})^{1/2}∥x∥_{HS(H_{n})}∥y∥_{HS(H_{k})},

for all x ∈ P̂_{n}ℂ[𝔾̂] and y ∈ P̂_{k}ℂ[𝔾̂].

Of course if l ∈ ℕ is such that U^{l} is not a subrepresentation of U^{n}⊠U^{k}, then the inequality in Proposition 4.13 is trivially valid since the left-hand side is then zero. So to proceed towards proving Proposition 4.13, let us fix an inclusion U^{l} ⊂ U^{n} ⊠ U^{k}. Recall that we defined in equation (3.2) a morphism ρ_{l}^{n⊠k} ∈ Mor(U^{l}, U^{n} ⊠ U^{k}) ⊂ TL_{2l,2n+2k}(δ) which has the property that C_{(n,k,l)}^{−1/2}ρ_{l}^{n⊠k}

is an isometry from H_{l} onto Δ̂(P̂_{l})(H_{n} ⊗ H_{k}), where C_{(n,k,l)} is the constant defined in (3.3). We will use the morphism ρ_{l}^{n⊠k} to obtain the required norm estimate in Proposition 4.13. To do this, we first need the following lemma which gives a uniform estimate on the size of C_{(n,k,l)}.

Lemma 4.14. With the notation above, there exists a constant D_{0} > 0 (independent of n, k, l and only depending on dim B) such that D_{0} ≤ C_{(n,k,l)} ≤ 1.

Proof. Since ρ_{l}^{n⊠k} is a contraction, the upper bound C_{(n,k,l)} ≤ 1 is immediate. The lower bound can be obtained by analyzing the expansion of C_{(n,k,l)} in terms of the q-numbers. This, however, has already been done by Vaes and Vergnioux in [36], Lemma A.6. Note that in [36], the authors work with a concrete representation of the Temperley-Lieb algebras which is generally different from the one considered here. However, since all representations under consideration are faithful, their estimates transfer directly to our context.

Proof of Proposition 4.13. With U^{l} ⊂ U^{n} ⊠ U^{k} fixed, let 0 ≤ r ≤ 2 min{n, k} be such that l = k + n − r. From Lemma 4.14 and the discussion preceding it, it follows that

∥Δ̂(P̂_{l})(x ⊗ y)Δ̂(P̂_{l})∥_{HS(H_{n}⊗H_{k})} = C_{(n,k,l)}^{−1}∥(ρ_{l}^{n⊠k})^{∗}(x ⊗ y)ρ_{l}^{n⊠k}∥_{HS(H_{l})}
≤ D_{0}^{−1}∥(ρ_{l}^{n⊠k})^{∗}(x ⊗ y)ρ_{l}^{n⊠k}∥_{HS(H_{l})}.

We now consider two possible cases: either r is odd or even. Suppose first that r = 2s + 1 is odd. Identify H_{l} with the highest weight subspace Δ̂(P̂_{l})(H_{n−s−1}⊗H_{k−s}), and H_{n}, H_{k} with the highest weight subspaces of H_{n−s−1} ⊗ H_{1}^{⊗(s+1)} and H_{1}^{⊗s} ⊗ H_{k−s}, respectively. Now recall Section 3.2 where we fixed an orthonormal basis {e_{i}}_{i=1}^{d_{1}} for H_{1}, a unitary F_{1} such that (F_{1} ⊗ 1)\overline{U^{1}}(F_{1}^{−1} ⊗ 1) = U^{1} and an isometric morphism t_{2} = d_{1}^{−1/2} ∑_{i} e_{i} ⊗ F_{1}e_{i} ∈ Mor(1, U^{1} ⊠ U^{1}). Let e_{ij} and f_{ij} denote the standard matrix units for B(H_{1}) relative to the orthonormal bases {e_{i}}_{i} and {f_{i} = F_{1}e_{i}}_{i}, respectively. Given functions i, j : [s] → [d_{1}], we will also write e_{i} and f_{i} for the tensors e_{i(1)} ⊗ … e_{i(s)} and f_{i(1)} ⊗ … ⊗ f_{i(s)} ∈ H_{1}^{⊗s}, respectively, and write e_{ij} and f_{ij} for the matrix units e_{i(1)j(1)}⊗…⊗e_{i(s)j(s)} and f_{i(1)j(1)}⊗…⊗f_{i(s)j(s)} ∈ B(H_{1}^{⊗s}), respectively. With these identifications, we can uniquely express

x = ∑_{\begin{matrix}1≤i_{0},j_{0}≤d_{1} \\ i,j:[s]→[d_{1}] \end{matrix}} x_{i_{0},i,j_{0},j} ⊗ e_{i_{0}j_{0}} ⊗ e_{ij}, y = ∑_{u,v:[s]→[d_{1}]} f_{uv} ⊗ y_{u,v},

where {x_{i_{0},i,j_{0},j}}_{i_{0},i,j_{0},j} ⊂ B(H_{n−s−1}) and {y_{u,v}}_{u,v} ⊂ B(H_{k−s}). Note that ∥x∥_{HS(H_{n})}^{2} = ∑_{i,j,i_{0},j_{0}} ∥x_{i_{0},i,j_{0},j}∥_{HS(H_{n−s−1})}^{2} and ∥y∥_{HS(H_{k})}^{2} = ∑_{u,v} ∥y_{u,v}∥_{HS(H_{k−s})}^{2}. By repeatedly using the expression (3.7) for t_{2} in the re-

cursion formula (3.5), it follows that [2s+1]_{q}^{1/2}t_{2s} = (p_{2s}⊗p_{2s}) ∑_{i:[s]→[d_{1}]} e_{i}⊗fǐ, where for any multi-index i = (i(1), …, i(s − 1), i(s)), ǐ := (i(s), i(s − 1), …, i(1)). Plugging this expression for t_{2s} into (3.6), we obtain

1_{1} ⊗ t_{2s+1} ⊗ 1_{1}

= [2s + 2]_{q}^{−1/2}[2]_{q}^{−1/2}(1_{1} ⊗ p_{2s+1} ⊗ p_{2s+1} ⊗ 1_{1})(1_{2} ⊗ ∑_{i:[s]→[d_{1}]} e_{i} ⊗ fǐ ⊗ 1_{2})m^{∗},

which then yields

(ρ_{l}^{n⊠k})^{∗}(x ⊗ y)ρ_{l}^{n⊠k} = p_{2l}(1_{2n−2s−1} ⊗ t_{2s+1}^{∗} ⊗ 1_{2k−2s−1})

× ∑_{\begin{matrix}i,j,u,v:[s]→[d_{1}] \\ 1≤i_{0},j_{0}≤d_{1} \end{matrix}}(x_{i_{0},i,j_{0},j} ⊗ e_{i_{0}j_{0}} ⊗ e_{ij} ⊗ f_{uv} ⊗ y_{u,v})

× (1_{2n−2s−1} ⊗ t_{2s+1} ⊗ 1_{2k−2s−1})p_{2l}

= [2]_{q}^{−1}[2s + 2]_{q}^{−1}p_{2l}

× ∑_{\begin{matrix}i,j:[s]→[d_{1}] \\ 1≤i_{0},j_{0}≤d_{1} \end{matrix}} (x_{i_{0},i,j_{0},j} ⊗ (m ⊗ 1_{2k−2s−2})(e_{i_{0}j_{0}} ⊗ y_{ǐ,ǰ})(m^{∗} ⊗ 1_{2k−2s−2}))p_{2l}

= [2]_{q}[2s + 2]_{q}^{−1}p_{2l} ∑_{\begin{matrix}i,j:[s]→[d_{1}] \\ 1≤i_{0},j_{0}≤d_{1} \end{matrix}} (x_{i_{0},i,j_{0},j} ⊗ (ρ_{k−s}^{1⊠(k−s)})^{∗}(e_{i_{0}j_{0}} ⊗ y_{ǐ,ǰ})ρ_{k−s}^{1⊠(k−s)})p_{2l}.

Note that in the last equality above, we have also used that (p_{2} ⊗p_{2k−2s})(m^{∗} ⊗ 1_{2k−2s−2})p_{2k−2s} = [2]_{q}ρ_{k−s}^{1⊠(k−s)}, which follows from (3.2). If we now take the Hilbert-Schmidt norm of the above expression and use the triangle and Cauchy-Schwarz inequalities, we obtain

∥(ρ_{l}^{n⊠k})^{∗}(x ⊗ y)ρ_{l}^{n⊠k}∥_{HS(H_{l})}

≤ \frac{[2]_{q}}{[2s + 2]_{q}}

× ∥ ∑_{\begin{matrix}i,j:[s]→[d_{1}] \\ 1≤i_{0},j_{0}≤d_{1} \end{matrix}} x_{i_{0},i,j_{0},j} ⊗ (ρ_{k−s}^{1⊠(k−s)})^{∗}(e_{i_{0}j_{0}} ⊗ y_{ǐ,ǰ})ρ_{k−s}^{1⊠(k−s)})∥_{HS(H_{n−s−1}⊗H_{k−s})}

≤ [2]_{q}[2s + 2]_{q}^{−1}( ∑_{i,j,i_{0},j_{0}} ∥x_{i_{0},i,j_{0},j}∥_{HS(H_{n−s−1})}^{2})^{1/2}

× ( ∑_{i,j,i_{0},j_{0}} ∥(ρ_{k−s}^{1⊠(k−s)})^{∗}(e_{i_{0}j_{0}} ⊗ y_{ǐ,ǰ})ρ_{k−s}^{1⊠(k−s)}∥_{HS(H_{k−s})}^{2})^{1/2}

= [2]_{q}[2s + 2]_{q}^{−1}∥x∥_{HS(H_{n})}

× C_{(1,k−s,k−s)}( ∑_{i,j,i_{0},j_{0}} ∥Δ̂(P̂_{k−s})(e_{i_{0}j_{0}} ⊗ y_{ǐ,ǰ})Δ̂(P̂_{k−s})∥_{HS(H_{1}⊗H_{k−s})}^{2})^{1/2}

≤ [2]_{q}[2s + 2]_{q}^{−1}∥x∥_{HS(H_{n})}( ∑_{i,j,i_{0},j_{0}} ∥e_{i_{0}j_{0}} ⊗ y_{ǐ,ǰ}∥_{HS(H_{1}⊗H_{k−s})}^{2})^{1/2}

= [2]_{q}[2s + 2]_{q}^{−1}∥x∥_{HS(H_{n})}( ∑_{i,j,i_{0},j_{0}} ∥y_{ǐ,ǰ}∥_{HS(H_{k−s})}^{2})^{1/2}

= [2]_{q}[3]_{q}[2s + 2]_{q}^{−1}∥x∥_{HS(H_{n})}∥y∥_{HS(H_{k})}.

Now consider the second case where r = 2s is even. We will only sketch this situation, as it is similar to (and in fact simpler than) the previous case. We now identify H_{l} with the highest weight subspace Δ̂(P̂_{l})(H_{n−s} ⊗ H_{k−s}), and H_{n}, H_{k} with the highest weight subspaces of H_{n−s} ⊗ H_{1}^{⊗s} and H_{1}^{⊗s} ⊗ H_{k−s}, respectively. Using the notation from the previous case, write

x = ∑_{i,j:[s]→[d_{1}]} x_{i,j} ⊗ e_{ij}, y = ∑_{u,v:[s]→[d_{1}]} f_{uv} ⊗ y_{u,v},

where {x_{i,j}}_{i,j} ⊂ B(H_{n−s}) and {y_{u,v}}_{u,v} ⊂ B(H_{k−s}). Since [2s + 1]_{q}^{1/2}t_{2s} = (p_{2s} ⊗ p_{2s}) ∑_{i:[s]→[d_{1}]} e_{i} ⊗ fǐ, it is easy to verify that

(ρ_{l}^{n⊠k})^{∗}(x ⊗ y)ρ_{l}^{n⊠k}
= p_{2l}(1_{2n−2s} ⊗ t_{2s}^{∗} ⊗ 1_{2k−2s})

× ( ∑_{i,j,u,v:[s]→[d_{1}]} x_{i,j} ⊗ e_{ij} ⊗ f_{uv} ⊗ y_{u,v})(1_{2n−2s} ⊗ t_{2s} ⊗ 1_{2k−2s})p_{2l}

= [2s + 1]^{−1}p_{2l}( ∑_{i,j:[s]→[d_{1}]} x_{i,j} ⊗ y_{ǐ,ǰ})p_{2l}.

Taking the Hilbert-Schmidt norm then yields

∥(ρ_{l}^{n⊠k})^{∗}(x ⊗ y)ρ_{l}^{n⊠k}∥_{HS(H_{l})} ≤ [2s + 1]_{q}^{−1}∥x∥_{HS(H_{n})}∥y∥_{HS(H_{k})}.

From the preceding two estimates we finally obtain the inequality

∥Δ̂(P̂_{l})(x ⊗ y)Δ̂(P̂_{l})∥_{HS(H_{n}⊗H_{k})} ≤ [r + 1]_{q}^{−1}[2]_{q}[3]_{q}D_{0}^{−1}∥x∥_{HS(H_{n})}∥y∥_{HS(H_{k})},

where l = n + k − r. To complete the proof, we therefore need to show that there exist constants 0 < D′ ≤ D′′ (depending only on q = q(dim B)) such that

D′ ≤ [r + 1]_{q}^{2}d_{l}d_{n}^{−1}d_{k}^{−1} = \frac{[r + 1]_{q}^{2}[2n + 2k − 2r + 1]_{q}}{[2n + 1]_{q}[2k + 1]_{q}} ≤ D′′.

But since

\frac{[r + 1]_{q}^{2}[2n + 2k − 2r + 1]_{q}}{[2n + 1]_{q}[2k + 1]_{q}} = \frac{(1 − q^{2r+2})^{2}(1 − q^{2(2n+2k−2r)+2})}{(1 − q^{2})(1 − q^{4n+2})(1 − q^{4k+2})},

the existence of the constants D′, D′′ is clear.

4.3 Further Approximation Properties

We will now use the property of rapid decay and the Haagerup property for L^{∞}(𝔾) to prove some additional approximation properties for L^{∞}(𝔾), C_{r}(𝔾) and the convolution algebra L^{1}(𝔾). In this section we will assume that dim B ≥ 5 (i.e., 𝔾 is not co-amenable), so that L^{∞}(𝔾) is non-injective, C_{r}(𝔾) is nonnuclear, and L^{1}(𝔾) does not have a bounded approximate identity (see [11], Theorem 1.1 and [12], Theorem 3.1). Recall that a Banach space X has the metric approximation property if there exists a net of finite rank contractions {Φ_{t} : X → X}_{t∈Λ} such that lim_{t} Φ_{t} = id_{X} pointwise in norm. If X is a dual Banach space, we say that X has the weak^{∗} metric approximation property if there is a net of weak^{∗} continuous finite rank contractions {Φ_{t} : X → X}_{t∈Λ} such that lim_{t} Φ_{t} = id_{X} pointwise in the weak^{∗} topology. Finally, recall that the left multiplier norm on L^{1}(𝔾) is the norm

∥ω∥_{M_{l}(L^{1}(𝔾))} := sup{∥ω ∗ ω′∥_{L^{1}(𝔾)} : ∥ω′∥_{L^{1}(𝔾)} = 1} (ω ∈ L^{1}(𝔾)),

where ∗ := (Δ_{r})_{∗} denotes the convolution product on L^{1}(𝔾). Obviously ∥ω∥_{M_{l}(L^{1}(𝔾))} = ∥L_{ω}∥_{B(L^{1}(𝔾))} ≤ ∥ω∥_{L^{1}(𝔾)}, where L_{ω} is the left-multiplication operator induced by ω ∈ L^{1}(𝔾).

Theorem 4.15. Let ψ be the δ-trace on a finite dimensional C^{∗}-algebra B and let 𝔾 = 𝔾_{aut}(B, ψ). Then the following approximation properties hold for the quantum groups 𝔾.

1. C_{r}(𝔾) has the metric approximation property.

2. L^{1}(𝔾) has a central approximate identity {ω_{n}}_{n≥1} such that sup_{n} ∥ω_{n}∥_{M_{l}(L^{1}(𝔾))} ≤ 1.

3. L^{∞}(𝔾) has the weak^{∗} metric approximation property.

Our proof of this theorem uses standard truncation techniques for completely positive maps using property RD. Compare with [24] and [14].

Proof of (1). We will use the notation and results from Theorem 4.2 and its proof. Let {Φ_{t}}_{t∈[t_{0},dim B)} ⊂ CB(L^{∞}(𝔾)) be the net of normal unital completely positive maps in (4.3), whose L^{2}-extensions are given by

Φ̂_{t} = ∑_{k∈ℕ} \frac{Π_{k}(t)}{Π_{k}(dim B)}P_{k}.

Recall that by Theorem 4.5, Φ_{t} restricts to a unital completely positive map on C_{r}(𝔾). Moreover, since lim_{t→dim B} \frac{Π_{k}(t)}{Π_{k}(dim B)} = 1 and ∥Φ_{t}∥_{B(C_{r}(𝔾))} = 1 for all t, it follows that lim_{t→dim B} Φ_{t} = id_{C_{r}(𝔾)} pointwise in norm. Now, for each n ∈ ℕ consider the finite rank truncation

Φ̂_{t,n} = ∑_{k≤n} \frac{Π_{k}(t)}{Π_{k}(dim B)}P_{k} ∈ B(L^{2}(𝔾)),

and let Φ_{t,n} : L^{∞}(𝔾) → C_{r}(𝔾) be the associated finite rank map determined by Φ̂_{t,n} ∘ Λ_{h} = Λ_{h} ∘ Φ_{t,n}. Using Theorem 4.10, we have ∥P_{k}∥_{B(C_{r}(𝔾))} ≤ ∥P_{k}∥_{L^{2}→L^{∞}} ≤ D(2k + 1) for each k ∈ ℕ. Therefore

∥Φ_{t} − Φ_{t,n}∥_{B(C_{r}(𝔾))} = ∥ ∑_{k≥n+1} \frac{Π_{k}(t)}{Π_{k}(dim B)}P_{k}∥_{B(C_{r}(𝔾))}

≤ ∑_{k≥n+1} \frac{Π_{k}(t)}{Π_{k}(dim B)}∥P_{k}∥_{B(C_{r}(𝔾))} ≤ D ∑_{k≥n+1} \frac{Π_{k}(t)}{Π_{k}(dim B)}(2k + 1)

≤ A(t_{0})D ∑_{k≥n+1}(2k + 1)(\frac{t}{dim B} )^{k} ⟶ 0 (n → ∞, t ∈ [t_{0}, dim B)),

where in the last line we have used (4.4). Consequently Φ_{t} = lim_{n→∞} Φ_{t,n} in operator norm and lim_{n→∞} ∥Φ_{t,n}∥ = ∥Φ_{t}∥ = 1. Therefore if we set Φ̃_{t,n} = ∥Φ_{t,n}∥^{−1}Φ_{t,n}, we obtain a family of finite rank contractions {Φ̃_{t,n}}_{t,n} which also satisfies lim_{n→∞} Φ̃_{t,n} = Φ_{t} in norm. Since we already know that lim_{t→dim B} Φ_{t} = id_{C_{r}(𝔾)} in the point-norm topology, we conclude that the set {Φ̃_{t,n}}_{t,n} contains id_{C_{r}(𝔾)} in its point-norm closure. We can now easily extract a sequence of finite rank contractions from {Φ̃_{t,n}}_{t,n} which yields the metric approximation property. Indeed, just choose any sequence of numbers {t(n)}_{n∈ℕ} ⊂ [t_{0}, dim B) such that

lim_{n→∞} t(n) = dim B and lim_{n→∞} ∑_{k≥n+1}(2k + 1)(\frac{t(n)}{dim B} )^{k} = 0.

Then the sequence {Ψ_{n}}_{n∈ℕ} ⊂ B(C_{r}(𝔾)) where Ψ_{n} = Φ̃_{t(n),n} does the job. Proof of (2). Identify L^{1}(𝔾) with the closure of C_{alg}(𝔾) with respect to the norm

∥ω∥_{L^{1}(𝔾)} := h(|ω|) = sup{|h(ωx)| : x ∈ C_{alg}(𝔾), ∥λ(x)∥_{L^{∞}(𝔾)} = 1}.

Consider the sequence {ω_{n}}_{n∈ℕ} ⊂ Z(L^{1}(𝔾)), where ω_{n} = ∥Φ_{t(n),n}∥^{−1} ∑_{k≤n} Π_{k}(t(n))χ_{k} and {t(n)}_{n∈ℕ} is the sequence chosen in the proof of part (1). Since lim_{n} t(n) = dim B, lim_{n} ∥Φ_{t(n),n}∥ = 1 and χ_{k} ∗ u_{ij}^{l} = u_{ij}^{l} ∗ χ_{k} = \frac{δ_{k,l}}{Π_{l}(dim B)}u_{ij}^{l} for each irreducible matrix element u_{ij}^{l} ∈ C_{alg}(𝔾) ⊂ L^{1}(𝔾), it follows that

L_{ω_{n}}ω = ω_{n} ∗ ω = ω ∗ ω_{n} ⟶ ω  (n → ∞, ω ∈ C_{alg}(𝔾) ⊂ L^{1}(𝔾)). (4.9)

To prove that (4.9) holds for each ω ∈ L^{1}(𝔾), the usual density argument shows that it is suffices to observe that ∥ω_{n}∥_{M_{l}(L^{1}(𝔾))} is uniformly bounded. This, however, is true because an easy duality calculation shows that (L_{ω_{n}})^{∗} = Ψ_{n} ∈ B(L^{∞}(𝔾)) and therefore ∥ω_{n}∥_{M_{l}(L^{1}(𝔾))} = ∥L_{ω_{n}}∥ = ∥Ψ_{n}∥_{B(L^{∞}(𝔾))} = ∥Ψ_{n}∥_{B(C_{r}(𝔾))} = 1. The result now follows. Proof of (3). Since the maps Ψ_{n} = (L_{ω_{n}})^{∗} are σ-weakly continuous finite rank contractions, and ∥L_{ω_{n}}ω − ω∥_{L^{1}(𝔾)} → 0 for all ω ∈ L^{1}(𝔾) = L^{∞}(𝔾)_{∗}, it follows by duality that lim_{n} Ψ_{n} = id_{L^{∞}(𝔾)} pointwise σ-weakly.

4.4 A Remark on Exactness

We close this section with a few remarks on the exactness of the reduced C^{∗}- algebras C_{r}(𝔾). We suspect that Corollary 4.17 below may already be known by the experts, but we could not find a reference. In [13], the notion of monoidal equivalence for compact quantum groups was introduced, and in [36], Vaes and Vergnioux showed that exactness of reduced C^{∗}-algebras of compact quantum groups is preserved by monoidal equivalence.

Theorem 4.16 ([36], Theorem 6.1). Let 𝔾 and 𝔾_{1} be monoidally equivalent compact quantum groups. Then C_{r}(𝔾) is exact if and only if C_{r}(𝔾_{1}) is exact.

Now let (B, ψ) be a finite dimensional C^{∗}-algebras equipped with a (possibly non-tracial) δ-form ψ : B → ℂ and consider the quantum automorphism group 𝔾 = 𝔾_{aut}(B, ψ). In [20], Section 9.3, it is shown that 𝔾 is monoidally equivalent to 𝔾_{1} = 𝔾_{aut}(M_{2}(ℂ), Tr(Q·)), where Q ∈ M_{2}(ℂ) is any positive invertible matrix satisfying Tr(Q^{−1}) = δ^{2}. Using [34], we can assume that 𝔾_{1} is isomorphic to SO_{q}(3) where 0 < q ≤ 1 is such that q + q^{−1} = δ. Since SO_{q}(3) is well known to be co-amenable, we have C_{r}(𝔾_{1}) ≅ C_{u}(𝔾_{1}) is nuclear (therefore exact) by [11], Theorem 1.1. This yields the following corollary.

Corollary 4.17. Let ψ be a δ-form on a finite dimensional C^{∗}-algebra B and let 𝔾 = 𝔾_{aut}(B, ψ). Then C_{r}(𝔾) is an exact C^{∗}-algebra.

5 Algebraic Structure

We now turn to some algebraic questions concerning the operator algebras L^{∞}(𝔾) and C_{r}(𝔾) associated to the trace-preserving quantum automorphism groups 𝔾 = 𝔾_{aut}(B, ψ). We will first prove a factoriality and fullness result for L^{∞}(𝔾), followed by a simplicity and uniqueness of trace result for C_{r}(𝔾). We conclude this section by studying property (AO)^{+} for L^{∞}(𝔾), and use this prove that L^{∞}(𝔾) is always a solid von Neumann algebra.

5.1 Factoriality and Fullness

Let (M, τ) be a II_{1}-factor (with unique faithful normal trace τ). Given a sequence {x_{n}}_{n=0}^{∞} ⊂ M, we say that {x_{n}}_{n=0}^{∞} is asymptotically central if ∥x_{n}y− yx_{n}∥_{2} → 0 for each y ∈ M. We say that {x_{n}}_{n=0}^{∞} is asymptotically trivial if ∥x_{n} − τ(x_{n})1∥_{2} → 0. Recall that M is said to be a full factor if every bounded asymptotically central sequence in M is asymptotically trivial. Concerning factoriality and fullness for L^{∞}(𝔾), we obtain the following result.

Theorem 5.1. Let ψ be the δ-trace on a finite dimensional C^{∗}-algebra B and let 𝔾 = 𝔾_{aut}(B, ψ). If dim B ≥ 8, then L^{∞}(𝔾) is a full type II_{1}-factor.

Remark 5.2. Note that there are six cases corresponding to dim B = 5, 6, 7 (i.e., B = C(X_{n}) with n = 5, 6, 7 and B = C(X_{n}) ⊕ M_{2}(ℂ) with n = 1, 2, 3) that are not addressed by Theorem 5.1. The question of factoriality remains open in these cases. See Remark 5.10 for more details. Note also that when dim B ≤ 4 and 𝔾 ≠ S_{4}^{+}, L^{∞}(𝔾) is commutative. If 𝔾 = S_{4}^{+}, the matrix model of Banica and Collins [7] (mentioned in Section 1) shows that L^{∞}(S_{4}^{+}) ↪ M_{4}(ℂ)\overline{⊗}L^{∞}(SU(2)) as an infinite dimensional subalgebra. Based on this explicit embedding, it seems likely that Z(L^{∞}(S_{4}^{+})) is infinite dimensional, but we are currently unable to verify this.

The arguments in our proof of Theorem 5.1 are based on the ideas of [36], Section 7, where factoriality for free orthogonal quantum groups is studied. A fundamental difference between the present situation and the one for free orthogonal quantum groups is that the fundamental representation of 𝔾 is reducible in our case. This means that this representation does not provide an optimal set of generators for L^{∞}(𝔾) with which to check commutators (i.e., factoriality). We remedy this by working instead with the matrix elements of the irreducible representation U^{1} of 𝔾 labeled by 1 ∈ ℕ. Recall that in Section 3.1 we fixed an orthonormal basis {e_{i}}_{i=1}^{d_{1}} for the Hilbert space H_{1}, with respect to which the irreducible representation U^{1} ∈ B(H_{1}) ⊗ C_{u}(𝔾) can be written as U^{1} = [u_{ij}^{1}]. Observe that by Theorem 3.8, every irreducible representation of 𝔾 is contained in some tensor power of U^{1} and therefore L^{∞}(𝔾) = {λ(u_{ij}^{1}) : 1 ≤ i, j ≤ d_{1}}′′. To study factoriality and fullness, we therefore consider the bounded linear map

T : L^{2}(𝔾) → H_{1} ⊗ L^{2}(𝔾) ⊗ H_{1}
T (yξ_{0}) = ∑_{i,j=1}^{d_{1}} F_{1}e_{j} ⊗ (λ(u_{ij}^{1})yξ_{0} − yλ(u_{ij}^{1})ξ_{0}) ⊗ e_{i}  (y ∈ L^{∞}(𝔾)),

where ξ_{0} = Λ_{h}(1) ∈ L^{2}(𝔾) is the cyclic and separating vector for L^{∞}(𝔾) and F_{1} ∈ U(d_{1}) is the unitary matrix from (3.7). Remark 5.3. The connection between the operator T and the factoriality or fullness of L^{∞}(𝔾) is obvious: L^{∞}(𝔾) is a factor if and only if ker T = ℂξ_{0}. If T |_{ℂξ_{0}^{⊥}} is bounded below, then L^{∞}(𝔾) is moreover a full factor. Indeed, if C > 0 is such that ∥T ξ∥_{H_{1}⊗L^{2}(𝔾)⊗H_{1}} ≥ C∥ξ∥_{L^{2}(𝔾)} for all ξ ∈ ℂξ_{0}^{⊥} and {x_{n}}_{n=1}^{∞} ⊂ L^{∞}(𝔾) is an asymptotically central sequence, then

∥x_{n} − h(x_{n})1∥_{L^{2}(𝔾)} ≤ C^{−1}∥T (x_{n} − h(x_{n})1)∥_{H_{1}⊗L^{2}(𝔾)⊗H_{1}} ⟶ 0 (n → ∞).

5.2 Analysis of the operator T

The commutator operator T compares the left and right action of the operators {λ(u_{ij}^{1})}_{1≤i,j≤d_{1}} on L^{2}(𝔾). At the level of representation theory, this can be interpreted in terms of comparing certain decompositions of tensor products of irreducible representations. This leads us to consider, for each k ≥ 1 and α ∈ {0, ±1}, the isometric morphisms ϕ_{k,L}^{(α)} : H_{k} → H_{1} ⊗ H_{k+α} and ϕ_{k,R}^{(α)} : H_{k} → H_{k+α} ⊗ H_{1} defined using formulas (3.2)–(3.3) as follows:

ϕ_{k,L}^{(+1)} = C_{(1,k+1,k)}^{−1/2}ρ_{k}^{1⊠(k+1)} = (\frac{[3]_{q}[2k + 1]_{q}}{[2k + 3]_{q}} )^{1/2}(p_{2} ⊗ p_{2k+2})(t_{2} ⊗ 1_{2k})p_{2k},
ϕ_{k,R}^{(+1)} = C_{(k+1,1,k)}^{−1/2}ρ_{k}^{(k+1)⊠1} = (\frac{[3]_{q}[2k + 1]_{q}}{[2k + 3]_{q}} )^{1/2}(p_{2k+2} ⊗ p_{2})(1_{2k} ⊗ t_{2})p_{2k},
ϕ_{k,L}^{(0)} = C_{(1,k,k)}^{−1/2}ρ_{k}^{1⊠k} = (\frac{[2k]_{q}}{[2k + 2]_{q}} )^{1/2}(p_{2} ⊗ p_{2k})(m^{∗} ⊗ 1_{2k−2})p_{2k},
ϕ_{k,R}^{(0)} = C_{(k,1,k)}^{−1/2}ρ_{k}^{k⊠1} = (\frac{[2k]_{q}}{[2k + 2]_{q}} )^{1/2}(p_{2k} ⊗ p_{2})(1_{2k−2} ⊗ m^{∗})p_{2k},
ϕ_{k,L}^{(−1)} = C_{(1,k−1,k)}^{−1/2}ρ_{k}^{1⊠(k−1)} = (p_{2} ⊗ p_{2k−2})p_{2k},
ϕ_{k,R}^{(−1)} = C_{(k−1,k,k)}^{−1/2}ρ_{k}^{(k−1)⊠1} = (p_{2k−2} ⊗ p_{2})p_{2k}.

The following theorem shows that, relative to the decomposition L^{2}(𝔾) ≅ ⨁_{k∈ℕ} H_{k} ⊗ H_{k} given by (2.3)–(2.4), the commutator operator T is blocktridiagonal and built quite naturally from the above list of isometric morphisms.

Theorem 5.4. There is a decomposition T = T^{(+1)} + T^{(0)} + T^{(−1)}, where for each α ∈ {0, ±1},

T^{(α)}ξ_{0} = 0,  T^{(α)}(H_{k} ⊗ H_{k}) ⊆ H_{1} ⊗ H_{k+α} ⊗ H_{k+α} ⊗ H_{1}  (k ≥ 1),

and

T^{(+1)}|_{H_{k}⊗H_{k}} = (\frac{[2k + 3]_{q}}{[2k + 1]_{q}} )^{1/2}(ϕ_{k,L}^{(+1)} ⊗ ϕ_{k,R}^{(+1)} − σϕ_{k,R}^{(+1)} ⊗ σ^{∗}ϕ_{k,L}^{(+1)}),
T^{(0)}|_{H_{k}⊗H_{k}} = ϕ_{k,L}^{(0)} ⊗ ϕ_{k,R}^{(0)} − σϕ_{k,R}^{(0)} ⊗ σ^{∗}ϕ_{k,L}^{(0)},
T^{(−1)}|_{H_{k}⊗H_{k}} = (\frac{[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}(ϕ_{k,L}^{(−1)} ⊗ ϕ_{k,R}^{(−1)} − σϕ_{k,R}^{(−1)} ⊗ σ^{∗}ϕ_{k,L}^{(−1)}),

where σ : H_{k} ⊗ H_{1} → H_{1} ⊗ H_{k} is the tensor flip map.

The proof of Theorem 5.4 is rather long and tedious, so we delay it to the Appendix (Section 6). Before commencing the proof of Theorem 5.1, we need one more remark. Remark 5.5. At various places during the proofs of Theorems 5.1 and 5.4, it will be useful to have slight modifications of the recursions (3.4)–(3.7) for the isometies t_{2k} ∈ Mor(1, U^{k} ⊠ U^{k}). Namely, we claim that t_{2} = [3]_{q}^{1/2}(1_{2} ⊗ p_{2})m^{∗}ν = [3]_{q}^{1/2}(p_{2} ⊗ 1_{2})m^{∗}ν and more generally,

t_{2k} = (\frac{[2k − 1]_{q}[3]_{q}}{[2k + 1]_{q}} )^{1/2}(1_{2k} ⊗ p_{2k})(1_{2k−2} ⊗ t_{2} ⊗ 1_{2k−2})t_{2k−2}  (5.1)
= (\frac{[2k − 1]_{q}[3]_{q}}{[2k + 1]_{q}} )^{1/2}(p_{2k} ⊗ 1_{2k})(1_{2k−2} ⊗ t_{2} ⊗ 1_{2k−2})t_{2k−2}.  (5.2)

While the above equalities are perhaps “diagrammatically” obvious, we can nonetheless algebraically verify these claims as follows. If k ≥ 2, let t̃_{2k} ∈ Mor(1, U^{(k−1)} ⊠ U^{1} ⊠ U^{k}) and ˜ t̃_{2k} ∈ Mor(1, U^{k} ⊠ U^{1} ⊠ U^{(k−1)}) be the morphisms appearing on the right sides of (5.1) and (5.2), respectively. Since Mor(1, U^{(k−1)} ⊠ U^{1} ⊠ U^{k}) is one-dimensional and t_{2k} ∈ Mor(1, U^{k} ⊠ U^{k}) ⊆ Mor(1, U^{(k−1)} ⊠ U^{1} ⊠ U^{k}), we must have t̃_{2k} = zt_{2k} for some z ∈ ℂ. Moreover, since \overline{z} = t̃_{2k}^{∗}t_{2k} = t_{2k}^{∗}t_{2k} = 1, we must have z = 1. A similar argument gives ˜t̃_{2k} = t_{2k}. The case of k = 1 goes similarly. We will now use the tridiagonal decomposition of T given by Theorem 5.4 to show that if the dimension of B is sufficiently large, then T is bounded below on ℂξ_{0}^{⊥} ⊂ L^{2}(𝔾). By Remark 5.3, this will prove that L^{∞}(𝔾) is a full factor. To obtain this bound, we actually prove that T^{(+1)}|_{ℂξ_{0}^{⊥}} is bounded below by some constant which tends to ∞ as dim B → ∞, while (T^{(0)} + T^{(−1)})|_{ℂξ_{0}^{⊥}} is bounded above by a constant which is uniformly bounded as a function of dim B. An application of the triangle inequality will then show that T |_{ℂξ_{0}^{⊥}} is bounded below for dim B sufficiently large. Notation 5.6. Given ξ ∈ L^{2}(𝔾), denote by ξ_{k} the component of ξ belonging to the subspace L_{k}^{2}(𝔾) ≅ H_{k} ⊗ H_{k}, and let ξ^{0} = ξ − ⟨ξ_{0}|ξ⟩ξ_{0} be the orthogonal projection of ξ onto ℂξ_{0}^{⊥}. We then have T ξ = T ξ^{0} = T^{(+1)}ξ^{0} +(T^{(0)} +T^{(−1)})ξ^{0}. Similarly, given η ∈ H_{1} ⊗ L^{2}(𝔾) ⊗ H_{1}, let η_{k} be the component of η belonging to the subspace H_{1} ⊗ H_{k} ⊗ H_{k} ⊗ H_{1}.

Quantum Automorphism Groups 1381

We start by finding a uniform upper bound for the norm of (T^{(0)} + T^{(−1)})|_{ℂξ_{0}^{⊥}}. Lemma 5.7.

∥(T^{(0)} + T^{(−1)})|_{ℂξ_{0}^{⊥}} ∥ ≤ 2(1 + q).

Proof. Theorem 5.4 implies that for any ξ ∈ L^{2}(𝔾),

∥T^{(0)}ξ^{0}∥^{2} = ∑_{k=1}^{∞} ∥(T^{(0)}ξ^{0})_{k}∥^{2} = ∑_{k=1}^{∞} ∥T^{(0)}ξ_{k}∥^{2} ≤ ∑_{k=1}^{∞} ∥T^{(0)}|_{H_{k}⊗H_{k}} ∥^{2}∥ξ_{k}∥^{2}

≤ ∑_{k=1}^{∞} 4∥ξ_{k}∥^{2} = 4∥ξ^{0}∥^{2},

∥T^{(−1)}ξ^{0}∥^{2} = ∑_{k=0}^{∞} ∥(T^{(−1)}ξ^{0})_{k}∥^{2} = ∑_{k=1}^{∞} ∥T^{(−1)}ξ_{k}∥^{2} ≤ ∑_{k=1}^{∞} ∥T^{(−1)}|_{H_{k}⊗H_{k}} ∥^{2}∥ξ_{k}∥^{2}

≤ ∑_{k=1}^{∞} \frac{4[2k − 1]_{q}}{[2k + 1]_{q}} ∥ξ_{k}∥^{2} ≤ 4q^{2}∥ξ^{0}∥^{2},

where in the last line we have used the fact that

k ↦ \frac{[2k − 1]_{q}}{[2k + 1]_{q}} = q^{2}(\frac{1 − q^{4k−2}}{1 − q^{4k+2}} ) is increasing and sup_{k} \frac{[2k − 1]_{q}}{[2k + 1]_{q}} = q^{2}.

The result now follows from the triangle inequality. Next we deal with more difficult task of bounding T^{(+1)}|_{ℂξ_{0}^{⊥}} from below. Fix ξ ∈ L^{2}(𝔾). Then ∥T^{(+1)}ξ^{0}∥^{2} = ∑_{k=1}^{∞} ∥T^{(+1)}ξ_{k}∥^{2} and for each k ≥ 1, Theorem 5.4 gives

∥T^{(+1)}ξ_{k}∥^{2}

= \frac{[2k + 3]_{q}}{[2k + 1]_{q}} ∥(ϕ_{k,L}^{(+1)} ⊗ ϕ_{k,R}^{(+1)} − σϕ_{k,R}^{(+1)} ⊗ σ^{∗}ϕ_{k,L}^{(+1)})ξ_{k}∥^{2}

= \frac{[2k + 3]_{q}}{[2k + 1]_{q}} (∥(ϕ_{k,L}^{(+1)} ⊗ ϕ_{k,R}^{(+1)})ξ_{k}∥^{2} + ∥(σϕ_{k,R}^{(+1)} ⊗ σ^{∗}ϕ_{k,L}^{(+1)})ξ_{k}∥^{2}

− 2Re⟨(σϕ_{k,R}^{(+1)} ⊗ σ^{∗}ϕ_{k,L}^{(+1)})^{∗}(ϕ_{k,L}^{(+1)} ⊗ ϕ_{k,R}^{(+1)})ξ_{k}|ξ_{k}⟩)

= \frac{[2k + 3]_{q}}{[2k + 1]_{q}} (2∥ξ_{k}∥^{2} − 2Re⟨((ϕ_{k,R}^{(+1)})^{∗}σ^{∗}ϕ_{k,L}^{(+1)}) ⊗ ((ϕ_{k,L}^{(+1)})^{∗}σϕ_{k,R}^{(+1)})ξ_{k}|ξ_{k}⟩)

≥ \frac{[2k + 3]_{q}}{[2k + 1]_{q}} (2∥ξ_{k}∥^{2} − 2∥(ϕ_{k,L}^{(+1)})^{∗}σϕ_{k,R}^{(+1)}∥^{2}∥ξ_{k}∥^{2}).

Therefore

∥T^{(+1)}ξ^{0}∥^{2} ≥ 2 ∑_{k=1}^{∞} \frac{[2k + 3]_{q}}{[2k + 1]_{q}} (1 − ∥(ϕ_{k,L}^{(+1)})^{∗}σϕ_{k,R}^{(+1)}∥^{2})∥ξ_{k}∥^{2}
≥ 2 min_{k≥1} \{\frac{[2k + 3]_{q}}{[2k + 1]_{q}} (1 − ∥(ϕ_{k,L}^{(+1)})^{∗}σϕ_{k,R}^{(+1)}∥^{2})\}∥ξ^{0}∥^{2},  (5.3)

Documenta Mathematica 18 (2013) 1349–1402

showing that we need to find some useful upper bounds for the norms ∥(ϕ_{k,L}^{(+1)})^{∗}σϕ_{k,R}^{(+1)}∥, where

(ϕ_{k,L}^{(+1)})^{∗}σϕ_{k,R}^{(+1)}

= \frac{[3]_{q}[2k + 1]_{q}}{[2k + 3]_{q}} p_{2k}(t_{2}^{∗} ⊗ 1_{2k})(p_{2} ⊗ p_{2k+2})σ(1_{2k} ⊗ t_{2})p_{2k}  (k ≥ 1). (5.4)

As a first observation, note that the naïve estimate ∥(ϕ_{k,L}^{(+1)})^{∗}σϕ_{k,R}^{(+1)}∥ ≤

\frac{[3]_{q}[2k+1]_{q}}{[2k+3]_{q}} is of no use since the right hand side of this inequality always exceeds 1. To obtain a sharper estimate, we need to unravel the right-hand side of (5.4) using the recursive structure of the Jones-Wenzl projections {p_{y}}_{y∈ℕ}. We will do this by first finding a suitable two-step recursion formula which expresses the projection p_{2k+2} in terms of the projection p_{2k}. Plugging this recursion into the right-hand side of (5.4) will yield an expression whose norm can be more sharply bounded from above. The required two-step recursion for p_{2k+2} will be obtained by iterating twice the following (one-step) recursion for the Jones-Wenzl projections {p_{y}}_{y∈ℕ} due to Frenkel and Khovanov ([21], Equation 3.8):

p_{y} = (1_{y} − ∑_{r=1}^{y−1}(−1)^{y−r−1} \frac{[2]_{q}[r]_{q}}{[y]_{q}} (1_{r−1} ⊗ t_{1} ⊗ 1_{y−r−1} ⊗ t_{1}^{∗}))
× (p_{y−1} ⊗ 1_{1})  (y ≥ 2).  (5.5)

Setting y = 2k + 2 in (5.5) and then applying (5.5) again to the resulting p_{2k+1} term yields

p_{2k+2} = (1_{2k+2} − ∑_{r=1}^{2k+1}(−1)^{2k+1−r} \frac{[2]_{q}[r]_{q}}{[2k + 2]_{q}} (1_{r−1} ⊗ t_{1} ⊗ 1_{2k+1−r} ⊗ t_{1}^{∗}))

× (1_{2k+2} − ∑_{l=1}^{2k}(−1)^{2k−l} \frac{[2]_{q}[l]_{q}}{[2k + 1]_{q}} (1_{l−1} ⊗ t_{1} ⊗ 1_{2k−l} ⊗ t_{1}^{∗} ⊗ 1_{1}))

× (p_{2k} ⊗ 1_{2}).

Multiplying this expression by 1_{2} ⊗ p_{2k} on the left, observing that p_{2k+2} = (1_{2} ⊗ p_{2k})p_{2k+2} and p_{2k}TL_{y,2k}(δ) = TL_{2k,y}(δ)p_{2k} = 0 for all y < 2k, we see that most terms in the above sums vanish. That is,

p_{2k+2} = (1_{2} ⊗ p_{2k})(1_{2k+2} − \frac{[2]_{q}}{[2k + 2]_{q}} (t_{1} ⊗ 1_{2k} ⊗ t_{1}^{∗})

+ \frac{[2]_{q}^{2}}{[2k + 2]_{q}} (1_{1} ⊗ t_{1} ⊗ 1_{2k−1} ⊗ t_{1}^{∗}))

× (1_{2k+2} − ∑_{l=1}^{2k}(−1)^{2k−l} \frac{[2]_{q}[l]_{q}}{[2k + 1]_{q}} (1_{l−1} ⊗ t_{1} ⊗ 1_{2k−L} ⊗ t_{1}^{∗} ⊗ 1_{1}))

× (p_{2k} ⊗ 1_{2})

= (1_{2} ⊗ p_{2k})[1_{2k+2} + \frac{[2]_{q}}{[2k + 1]_{q}} (t_{1} ⊗ 1_{2k−1} ⊗ t_{1}^{∗} ⊗ 1_{1})

− \frac{[2]_{q}^{2}}{[2k + 1]_{q}} (1_{1} ⊗ t_{1} ⊗ 1_{2k−2} ⊗ t_{1}^{∗} ⊗ 1_{1}) − \frac{[2]_{q}}{[2k + 2]_{q}} (t_{1} ⊗ 1_{2k} ⊗ t_{1}^{∗})

+ \frac{[2]_{q}[2k]_{q}}{[2k + 1]_{q}[2k + 2]_{q}} (t_{1} ⊗ 1_{2k−1} ⊗ t_{1}^{∗} ⊗ 1_{1})

+ \frac{[2]_{q}^{2}}{[2k + 2]_{q}} (1_{1} ⊗ t_{1} ⊗ 1_{2k−1} ⊗ t_{1}^{∗})

+ \frac{[2]_{q}^{3}}{[2k + 1]_{q}[2k + 2]_{q}} ((1_{1} ⊗ t_{1} ⊗ 1_{1})t_{1} ⊗ 1_{2k−2} ⊗ t_{1}^{∗}(1_{1} ⊗ t_{1}^{∗} ⊗ 1_{1}))

− \frac{[2]_{q}^{2}[2k]_{q}}{[2k + 1]_{q}[2k + 2]_{q}} (1_{1} ⊗ t_{1} ⊗ 1_{2k−2} ⊗ t_{1}^{∗} ⊗ 1_{1})](p_{2k} ⊗ 1_{2}).

After collecting like terms, this gives p_{2k+2}

= (1_{2} ⊗ p_{2k})[ \underbrace{1_{2k+2}}_{(a)} +(\frac{[2]_{q}}{[2k + 1]_{q}} + \frac{[2]_{q}[2k]_{q}}{[2k + 1]_{q}[2k + 2]_{q}} ) \underbrace{(t_{1} ⊗ 1_{2k−1} ⊗ t_{1}^{∗} ⊗ 1_{1})}_{(b)}

− (\frac{[2]_{q}^{2}}{[2k + 1]_{q}} + \frac{[2]_{q}^{2}[2k]_{q}}{[2k + 1]_{q}[2k + 2]_{q}} ) \underbrace{(1_{1} ⊗ t_{1} ⊗ 1_{2k−2} ⊗ t_{1}^{∗} ⊗ 1_{1})}_{(c)}

− \frac{[2]_{q}}{[2k + 2]_{q}} \underbrace{(t_{1} ⊗ 1_{2k} ⊗ t_{1}^{∗})}_{(d)} +\frac{[2]_{q}^{2}}{[2k + 2]_{q}} \underbrace{(1_{1} ⊗ t_{1} ⊗ 1_{2k−1} ⊗ t_{1}^{∗})}_{(e)}

+ \frac{[2]_{q}^{3}}{[2k + 1]_{q}[2k + 2]_{q}} \underbrace{((1_{1} ⊗ t_{1} ⊗ 1_{1})t_{1} ⊗ 1_{2k−2} ⊗ t_{1}^{∗}(1_{1} ⊗ t_{1}^{∗} ⊗ 1_{1}))}_{(f)} ]

× (p_{2k} ⊗ 1_{2}).

Documenta Mathematica 18 (2013) 1349–1402

If we now insert the above expression for p_{2k+2} into equation (5.4) and use the fact that p_{2}t_{1} = 0, we see that the contributions coming from (b), (d) and (e) are all zero. Thus,

(ϕ_{k,L}^{(+1)})^{∗}σϕ_{k,R}^{(+1)}

= \frac{[3]_{q}[2k + 1]_{q}}{[2k + 3]_{q}} [ \underbrace{p_{2k}(t_{2}^{∗} ⊗ p_{2k})σ(1_{2k} ⊗ t_{2})p_{2k}}_{:=α [resulting from (a)]}

− (\frac{[2]_{q}^{2}}{[2k + 1]_{q}} + \frac{[2]_{q}^{2}[2k]_{q}}{[2k + 1]_{q}[2k + 2]_{q}} )

× \underbrace{p_{2k}(t_{2}^{∗} ⊗ p_{2k})(1_{2} ⊗ (1_{1} ⊗ t_{1} ⊗ 1_{2k−2} ⊗ t_{1}^{∗} ⊗ 1_{1}))σ(1_{2k} ⊗ t_{2})p_{2k}}_{:=β [resulting from (c)]}

+ \frac{[2]_{q}^{3}}{[2k + 1]_{q}[2k + 2]_{q}} Ad(p_{2k})

× \underbrace{(t_{2}^{∗} ⊗ p_{2k})(1_{2} ⊗ ((1_{1} ⊗ t_{1} ⊗ 1_{1})t_{1} ⊗ 1_{2k−2} ⊗ t_{1}^{∗}(1_{1} ⊗ t_{1}^{∗} ⊗ 1_{1})))σ(1_{2k} ⊗ t_{2})}_{:=γ [resulting from (f)]} ]

= \frac{[2k + 1]_{q}}{[2k + 3]_{q}} p_{2k}σ^{∗}p_{2k} − (\frac{1 + \frac{[2k]_{q}}{[2k+2]_{q}}}{[2k + 3]_{q}} )p_{2k}(1_{2k−2} ⊗ m)σ^{∗}(m^{∗} ⊗ 1_{2k−2})p_{2k}

+ \frac{[2]_{q}}{[2k + 3]_{q}[2k + 2]_{q}} p_{2k}σp_{2k},

where, to obtain the last equality, we have substituted the following three identities for the quantities α, β and γ:

[3]_{q}α = p_{2k}( ∑_{i=1}^{d_{1}} e_{i}^{∗} ⊗ (F_{1}e_{i})^{∗} ⊗ p_{2k})( ∑_{j=1}^{d_{1}} e_{j} ⊗ p_{2k} ⊗ F_{1}e_{j})

= p_{2k} ∑_{i=1}^{d_{1}}((F_{1}e_{i})^{∗} ⊗ p_{2k})(p_{2k} ⊗ F_{1}e_{i}) = p_{2k}σ^{∗}p_{2k},

[2]_{q}^{2}[3]_{q}β = [3]_{q}p_{2k}(t_{2}^{∗} ⊗ p_{2k})(1_{2} ⊗ m^{∗} ⊗ 1_{2k−4} ⊗ m)σ(1_{2k} ⊗ t_{2})p_{2k}

(using (3.6))

= p_{2k}( ∑_{i=1}^{d_{1}} e_{i}^{∗} ⊗ (F_{1}e_{i})^{∗} ⊗ p_{2k})(1_{2} ⊗ m^{∗} ⊗ 1_{2k−4} ⊗ m)

× ( ∑_{j=1}^{d_{1}} e_{j} ⊗ p_{2k} ⊗ F_{1}e_{j})

= p_{2k} ∑_{i=1}^{d_{1}}((F_{1}e_{i})^{∗} ⊗ p_{2k})(1_{2k} ⊗ m)((m^{∗} ⊗ 1_{2k−2})p_{2k} ⊗ F_{1}e_{i})

= p_{2k}(1_{2k−2} ⊗ m) ∑_{i=1}^{d_{1}}((F_{1}e_{i})^{∗} ⊗ 1_{2k+2})((m^{∗} ⊗ 1_{2k−2})p_{2k} ⊗ F_{1}e_{i})

= p_{2k}(1_{2k−2} ⊗ m)σ^{∗}(m^{∗} ⊗ 1_{2k−2})p_{2k},

and

[3]_{q}[2]_{q}^{2}γ = [3]_{q}^{2}p_{2k}(t_{2}^{∗} ⊗ p_{2k})(1_{2} ⊗ t_{2} ⊗ 1_{2k−2} ⊗ t_{2}^{∗})σ(1_{2k} ⊗ t_{2})p_{2k}
(using the fact that t_{2} = [3]_{q}^{−1/2}[2]_{q}(p_{2} ⊗ p_{2})(1_{1} ⊗ t_{1} ⊗ 1_{1})t_{1})

= [3]_{q}^{2}p_{2k}((t_{2}^{∗} ⊗ 1_{2})(1_{2} ⊗ t_{2}) ⊗ 1_{2k−2} ⊗ t_{2}^{∗})σ(1_{2k} ⊗ t_{2})p_{2k}

= [3]_{q}p_{2k}(p_{2} ⊗ 1_{2k−2} ⊗ t_{2}^{∗})σ(1_{2k} ⊗ t_{2})p_{2k}

(since (t_{2}^{∗} ⊗ 1_{2})(1_{2} ⊗ t_{2}) = [3]_{q}^{−1}p_{2})

= p_{2k}(p_{2} ⊗ 1_{2k−2} ⊗ ∑_{i=1}^{d_{1}} e_{i}^{∗} ⊗ (F_{1}e_{i})^{∗})( ∑_{j=1}^{d_{1}} e_{j} ⊗ p_{2k} ⊗ F_{1}e_{j})

= p_{2k}(p_{2} ⊗ 1_{2k−2}) ∑_{i=1}^{d_{1}}(1_{2k} ⊗ e_{i}^{∗})(e_{i} ⊗ p_{2k}) = p_{2k}σp_{2k}.

A simple application of the triangle inequality using the fact that ∥m∥ = δ = [2]_{q} now yields our desired norm estimate.

Lemma 5.8. For each k ≥ 1,

∥(ϕ_{k,L}^{(+1)})^{∗}σϕ_{k,R}^{(+1)}∥ ≤ \frac{[2k + 1]_{q} + [2]_{q}^{2}(1 + \frac{[2k]_{q}}{[2k+2]_{q}} ) + \frac{[2]_{q}}{[2k+2]_{q}}}{[2k + 3]_{q}} .  (5.6)

Returning to inequality (5.3) with Lemma 5.8 now at hand, we obtain the lower bound ∥T^{(+1)}ξ^{0}∥^{2}

Documenta Mathematica 18 (2013) 1349–1402

≥ 2∥ξ^{0}∥^{2} min_{k≥1} \{\frac{[2k + 3]_{q}}{[2k + 1]_{q}} − \frac{([2k + 1]_{q} + [2]_{q}^{2}(1 + \frac{[2k]_{q}}{[2k+2]_{q}} ) + \frac{[2]_{q}}{[2k+2]_{q}} )^{2}}{[2k + 3]_{q}[2k + 1]_{q}} \}.

By expanding the square in the previous line and using the fact that min_{k} \frac{[2k+3]_{q}}{[2k+1]_{q}} = q^{−2}, max_{k} \frac{[2k+1]_{q}}{[2k+3]_{q}} = q^{2} and that y ↦ [y]_{q} is increasing, we finally obtain

∥T^{(+1)}ξ^{0}∥^{2} ≥ 2(q^{−2} − q^{2} − \frac{[2]_{q}^{4}(1 + q^{2})^{2}}{[3]_{q}[5]_{q}} − \frac{[2]_{q}^{2}}{[4]_{q}^{2}[3]_{q}[5]_{q}} − \frac{2[2]_{q}^{2}(1 + q^{2})}{[5]_{q}}

− \frac{2[2]_{q}}{[4]_{q}[5]_{q}} − \frac{2[2]_{q}^{3}(1 + q^{2})}{[4]_{q}[3]_{q}[5]_{q}})∥ξ^{0}∥^{2} =: C(q)∥ξ^{0}∥^{2},

If we now combine the preceding inequality with Lemma 5.7, we have

∥T ξ∥ = ∥T^{(+1)}ξ^{0} + (T^{(0)} + T^{(−1)})ξ^{0}∥ ≥ ∥T^{(+1)}ξ^{0}∥ − ∥(T^{(0)} + T^{(−1)})ξ^{0}∥
≥ (C(q)^{1/2} − 2(1 + q))∥ξ^{0}∥  (ξ ∈ L^{2}(𝔾)).  (5.7)

We are now ready to prove Theorem 5.1. The remaining ingredient we need is the following lemma, which will also be useful when studying the simplicity of C_{r}(𝔾) in Section 5.3. Recall that 0 < q < 1 was defined so that δ = \sqrt{dim B} = q + q^{−1}. Note that q = q(δ) = \frac{δ−\sqrt{δ^{2}−4}}{2} is a decreasing function of δ.

Lemma 5.9. Consider the function f(δ) := [3]_{q(δ)}^{−1/2}(C(q(δ))^{1/2} − 2(1 + q(δ))). Then f is an increasing function on the interval [\sqrt{8}, ∞) and f(\sqrt{8}) > 0.

Proof. From the definition of C(q), it follows that f(δ) = g(q(δ)) where

g(q) = \sqrt{2}[\frac{q^{−2}}{[3]_{q}} − \frac{q^{2}}{[3]_{q}} − \frac{[2]_{q}^{4}(1 + q^{2})^{2}}{[3]_{q}^{2}[5]_{q}} − \frac{[2]_{q}^{2}}{[4]_{q}^{2}[3]_{q}^{2}[5]_{q}} − \frac{2[2]_{q}^{2}(1 + q^{2})}{[3]_{q}[5]_{q}}

− \frac{2[2]_{q}}{[3]_{q}[4]_{q}[5]_{q}} − \frac{2[2]_{q}^{3}(1 + q^{2})}{[4]_{q}[3]_{q}^{2}[5]_{q}}]^{1/2} − \frac{2(1 + q)}{[3]_{q}^{1/2}}.

When δ = \sqrt{8}, q = \frac{\sqrt{8}−2}{2} ≈ 0.4142 and direct substitution yields f(δ) ≈ 0.1111 > 0. It is now just an elementary calculus exercise to verify that g is a decreasing function of q ∈ (0, \frac{\sqrt{8}−2}{2}] and therefore f is an increasing function of δ ∈ [\sqrt{8}, ∞).

Proof of Theorem 5.1. Since f(\sqrt{8}) > 0 and f is increasing by Lemma 5.9, inequality (5.7) shows that T |_{ℂξ_{0}^{⊥}} is bounded below by [3]_{q(δ)}^{1/2}f(δ) > 0 whenever dim B = δ^{2} ≥ 8. Thus L^{∞}(𝔾) is a full factor by Remark 5.3.

Remark 5.10. Note that if δ^{2} = 5, 6 or 7, one can check that [3]_{q(δ)}^{1/2}f(δ) ≤ −0.4386 < 0. So the preceding calculations yield no information about the factoriality or fullness of L^{∞}(𝔾) when 5 ≤ dim B ≤ 7.

5.3 Simplicity of C_{r}(𝔾)

We now consider the reduced C^{∗}-algebra C_{r}(𝔾) and show that, at least in most cases, it is simple with unique trace. To do this, we adapt the “conjugation by generators” method used in [36], Section 7. Write W^{1} = (id ⊗ λ)U^{1} where U^{1} is the irreducible representation of 𝔾 labeled by 1 ∈ ℕ and consider the unital completely positive map Φ : C_{r}(𝔾) → C_{r}(𝔾) defined by

Φ(x) = \frac{1}{2[3]_{q}} (Tr ⊗ id)((W^{1})^{∗}(1 ⊗ x)W^{1} + W^{1}(1 ⊗ x)(W^{1})^{∗}).

Observe that Φ is τ-preserving for any tracial state τ on C_{r}(𝔾) and that Φ(I) ⊆ I for any closed two-sided ideal I ⊆ C_{r}(𝔾). Moreover, a simple calculation shows that the L^{2}-extension of Φ, denoted by Φ̂, is given by

Φ̂Λ_{h}(a) = (id_{L^{2}(𝔾)} − \frac{1}{2[3]_{q}} T^{∗}T )Λ_{h}(a)  (a ∈ C_{u}(𝔾)),  (5.8)

where T : L^{2}(𝔾) → H_{1} ⊗ L^{2}(𝔾) ⊗ H_{1} is the commutator map introduced in Section 5.1. The key result of this section is the following proposition.

Proposition 5.11. Let 𝔾 = 𝔾_{aut}(B, ψ), where B is a finite dimensional C^{∗}- algebra with δ-trace ψ. If dim B ≥ 8, then

lim_{n→∞} Φ^{n}(x) = h(x)1_{C_{r}(𝔾)} (x ∈ C_{r}(𝔾)).

Proof. Note that it is sufficient to prove this result for x_{0} belonging to the dense Hopf ∗-subalgebra λ(C_{alg}(𝔾)) ⊂ C_{r}(𝔾). Now take a ∈ C_{alg}(𝔾) and consider x = λ(a). Since a ∈ C_{alg}(𝔾), there is an r ≥ 0 such that x ∈ ⨁_{k=0}^{r} λ(C_{k}(𝔾)), where C_{k}(𝔾) is the subspace of C_{alg}(𝔾) spanned by the matrix elements of the irreducible representation U^{k}. Now let n ∈ ℕ and consider Φ^{n}(x) − h(x)1 = Φ^{n}(x − h(x)1), which evidently belongs to ⨁_{k=1}^{r+2n} λ(C_{k}(𝔾)). Writing Φ^{n}(x − h(x)1) = ∑_{k=1}^{r+2n} λ(z_{k}) with z_{k} ∈ C_{k}(𝔾) and using property RD (Theorem 4.10), we obtain

∥Φ^{n}(x) − h(x)1∥_{C_{r}(𝔾)} ≤ ∑_{k=1}^{r+2n} ∥λ(z_{k})∥_{C_{r}(𝔾)} ≤ ∑_{k=1}^{r+2n}(2k + 1)∥Λ_{h}(z_{k})∥_{L^{2}(𝔾)}

≤ (2r + 4n + 1)\sqrt{r + 2n}( ∑_{k=1}^{r+2n} ∥Λ_{h}(z_{k})∥_{L^{2}(𝔾)}^{2})^{1/2}

= (2r + 4n + 1)\sqrt{r + 2n}∥Φ̂^{n}(Λ_{h}(a − h(a)1))∥_{L^{2}(𝔾)}

≤ (2r + 4n + 1)\sqrt{r + 2n}∥Φ̂|_{ℂξ_{0}^{⊥}}∥^{n}∥Λ_{h}(a − h(a)1)∥_{L^{2}(𝔾)}.

Now assume that δ = \sqrt{dim B} ≥ \sqrt{8}. From (5.7) and Lemma 5.9, it follows that T^{∗}T |_{ℂξ_{0}^{⊥}} ≥ [3]_{q}f(δ)^{2}id_{ℂξ_{0}^{⊥}} and therefore

Φ̂|_{ℂξ_{0}^{⊥}} = id_{ℂξ_{0}^{⊥}} − \frac{1}{2[3]_{q}} T^{∗}T |_{ℂξ_{0}^{⊥}} ≤ (1 − \frac{1}{2}f(δ)^{2})id_{ℂξ_{0}^{⊥}} ≤ (1 − \frac{1}{2}f(\sqrt{8})^{2})id_{ℂξ_{0}^{⊥}}.

Since f(\sqrt{8}) ≈ 0.1111, ∥Φ̂|_{ℂξ_{0}^{⊥}}∥ ≤ (1 − \frac{1}{2}f(\sqrt{8})^{2}) < 1, which gives

lim sup_{n→∞} ∥Φ^{n}(x) − h(x)1∥_{C_{r}(𝔾)}

≤ lim_{n→∞}(2r + 4n + 1)\sqrt{r + 2n}(1 − \frac{1}{2}f(\sqrt{8})^{2})^{n}∥Λ_{h}(a − h(a)1)∥_{L^{2}(𝔾)} = 0.

Simplicity and uniqueness of trace for C_{r}(𝔾) are now immediate consequences (as long as dim B ≥ 8).

Corollary 5.12. Let ψ be the δ-trace on a finite dimensional C^{∗}-algebra B and let 𝔾 = 𝔾_{aut}(B, ψ). If dim B ≥ 8, then C_{r}(𝔾) is simple with unique trace.

Proof. Let I ⊆ C_{r}(𝔾) be a non-zero closed two-sided ideal and let a ∈ I be such that h(a) ≠ 0. Then by Proposition 5.11, h(a)1 = lim_{n→∞} Φ^{n}(a) ∈ I. Therefore I = C_{r}(𝔾) and C_{r}(𝔾) is simple. Now let τ be any tracial state on C_{r}(𝔾). Since τ ∘ Φ^{n} = τ for all n, we have τ(x) = lim_{n→∞} τ(Φ^{n}(x)) = τ(h(x)1) = h(x) for all x ∈ C_{r}(𝔾).

5.4 Solidity

In this final section we show that the von Neumann algebras L^{∞}(𝔾) considered in this paper always have the Akemann-Ostrand property. Recall that a von Neumann M ⊆ B(H) has the Akemann-Ostrand property (or property (AO)) if there exist σ-weakly dense unital C^{∗}-subalgebras A ⊆ M and B ⊆ M′ such that A is locally reflexive and the ∗-homomorphism π∘m : A⊗_{alg}B → B(H)/K(H) is continuous with respect to the minimal tensor norm. Here m : A⊗_{alg}B → B(H) is the multiplication map and π : B(H) → B(H)/K(H) is the canonical quotient map. Note that an exact C^{∗}-algebra is always locally reflexive. We now state the main result of this section. Below we denote by J : L^{2}(𝔾) → L^{2}(𝔾) the modular conjugation operator for L^{∞}(𝔾) = λ(C_{u}(G))′′, so that JL^{∞}(𝔾)J = L^{∞}(𝔾)′.

Theorem 5.13. Let 𝔾 = 𝔾_{aut}(B, ψ), where B is a finite dimensional C^{∗}- algebra equipped with a δ-form ψ. Then there exists a unital completely positive map θ : C_{r}(𝔾) ⊗ JC_{r}(𝔾)J → B(L^{2}(𝔾)) such that

θ(a ⊗ JbJ) − aJbJ ∈ K(L^{2}(𝔾)) (a, b ∈ C_{r}(𝔾)).

In particular, L^{∞}(𝔾) has property (AO).

In [31], Ozawa proved that a finite von Neumann algebra M with property (AO) is solid. That is, for any diffuse von Neumann subalgebra A ⊂ M, the relative commutant A′ ∩ M is injective. Recall that a II_{1}-factor M is said to be prime if for every tensor product decomposition M ≅ M_{1}\overline{⊗}M_{2}, either M_{1} or M_{2} is finite dimensional. It is easy to see that a solid II_{1}-factor is prime unless it is injective. As a consequence of Corollary 4.17, Theorem 5.1 and the above discussion, the following result is immediate.

Corollary 5.14. Let ψ be the δ-trace on B and let 𝔾 = 𝔾_{aut}(B, ψ). Then L^{∞}(𝔾) is solid and it is a prime II_{1}-factor if dim B ≥ 8.

Remark 5.15. It is perhaps worthwhile pointing out that the local reflexivity of C_{r}(𝔾) is not actually required to establish the solidity of L^{∞}(𝔾). To see this, note that an examination of the proof that property (AO) implies solidity in [31] shows that local reflexivity is only needed to establish, for each finite dimensional operator system E ⊂ L^{∞}(𝔾), the existence of a net of unital completely positive maps {Ψ_{t}^{E} : E → C_{r}(𝔾)}_{t} converging pointwise σ-weakly to id_{E}. In our case, we can just take Ψ_{t}^{E} = Φ_{t}|_{E}, where {Φ_{t}}_{t<dim B} is the net constructed in the proof of Theorem 4.2. Indeed, a standard estimate using property (RD) (see for example [14], Proposition 5.3) shows that Φ_{t} is ultracontractive, that is Φ̂_{t}(L^{2}(𝔾)) ⊆ Λ_{h}(C_{r}(𝔾)) for all t. In particular, Φ_{t}(L^{∞}(𝔾)) ⊆ C_{r}(𝔾) for all t. Remark 5.16. The existence of a completely positive map θ in Theorem 5.13 such that π ∘ θ = π ∘ m imposes a structural property on L^{∞}(𝔾) which is stronger than property (AO). Using the terminology of Isono [25], Theorem 5.13 says that L^{∞}(𝔾) has property (AO)^{+}. Property (AO)^{+} can be regarded as a generalization to finite von Neumann algebras of the description of biexactness for discrete groups in terms of their reduced C^{∗}-algebras (see [15], Chapter 15). In [33], Popa and Vaes proved that the II_{1}-factors M = L(Γ) associated to i.c.c., weakly amenable, bi-exact groups Γ are strongly solid. That is, for any diffuse injective von Neumann subalgebra A ⊆ M, the von Neumann algebra generated by the normalizer of A, N_{M}(A) = {u ∈ U(M) : uAu^{∗} = A}, is injective. Recently, Isono [25] extended the results of [33] and proved that a finite von Neumann algebra with property (AO)^{+} and the weak^{∗}-completely bounded approximation property (W^{∗}-CBAP) is strongly solid. As a result, if one could prove that the von Neumann algebras L^{∞}(𝔾) considered here have the W^{∗}-CBAP, their strong solidity would be an immediate consequence. Based on recent work of Freslon [22] on the weak amenability free orthogonal quantum groups, we expect that L^{∞}(𝔾) always has this approximation property.

Proof of Theorem 5.13. Our arguments follow very closely those of Vergnioux [37], where property (AO)^{+} was established for the duals of universal discrete quantum groups. We note that the proof in [37] makes extensive use of the notion of a Cayley graph associated to a discrete quantum group 𝔾̂, and assumes that this graph is a tree. In the case of 𝔾 = 𝔾_{aut}(B, ψ) of interest here, the Cayley graph of 𝔾̂ is no longer a tree. In this case, since we have the fusion rules U^{1} ⊠ U^{k} ≅ U^{k} ⊠ U^{1} ≅ U^{k+1} ⊕ U^{k} ⊕ U^{k−1} (k ≥ 1), the Cayley graph is obtained by taking the infinite tree A_{∞} and adding a loop starting and ending at each vertex k ≠ 0 (resulting from the inclusions U^{k} ⊂ U^{1} ⊠ U^{k}, U^{k} ⊠ U^{1}). Since such a graph is not “too far” from being a tree, we can readily adapt the proof from [37] to our context. Below we freely use the notation from Sections 3.2, 4.2 and 5.1. We also assume that δ^{2} = q + q^{−1} ≥ 5, so that C_{r}(𝔾) is not nuclear.

For each l ∈ ℕ, put m_{l} = [2l + 1]_{q}, α_{l} = ( ∑_{n,k∈ℕ: n+k=l} \frac{m_{n}m_{k}}{m_{l}} )^{− \frac{1}{2}} and define a linear map

V : ℂ[𝔾̂] → ℂ[𝔾̂] ⊗_{alg} ℂ[𝔾̂]; V (x) = ∑_{l∈ℕ} α_{l} ∑_{n,k∈ℕ: n+k=l} Δ̂(P̂_{l}x)(P̂_{n} ⊗ P̂_{k}).

Using Lemma 4.12 and its proof, it follows that V extends to an isometry from ℓ^{2}(𝔾̂) → ℓ^{2}(𝔾̂) ⊗ ℓ^{2}(𝔾̂), and V^{∗}(P̂_{n} ⊗ P̂_{k})y = α_{n+k}P̂_{n+k}Conv((P̂_{n} ⊗ P̂_{k})y) (y ∈ ℓ^{2}(𝔾̂) ⊗ ℓ^{2}(𝔾̂), n, k ∈ ℕ). Identifying ℓ^{2}(𝔾̂) ≅ L^{2}(𝔾) via the Fourier transform F, we can form a unital completely positive map

θ : C_{r}(𝔾) ⊗ JC_{r}(𝔾)J → B(L^{2}(𝔾)); θ(x) = V^{∗}xV.

Our goal is to prove that

aJbJV^{∗} − V^{∗}(a ⊗ JbJ) ∈ K(L^{2}(𝔾) ⊗ L^{2}(𝔾), L^{2}(𝔾))  (5.9)

for all a, b ∈ C_{r}(𝔾), which will show that θ satisfies the hypotheses of the theorem. Note that since aJbJV^{∗} − V^{∗}(a ⊗ JbJ) = a(JbJV^{∗} − V^{∗}(1 ⊗ JbJ)) + (aV^{∗} − V^{∗}(a ⊗ 1))(1 ⊗ JbJ), it suffices to prove (5.9) when one of a or b equals 1. Let us start with b = 1. Note furthermore that we only need to prove (5.9) for a = λ(v) ∈ C_{r}(𝔾), where v ∈ C_{u}(𝔾) is a matrix element of the irreducible representation U^{1} = U ⊖ U^{0}. This follows because the norm-closed subalgebra of C_{r}(𝔾) generated by all such a = λ(v) equals C_{r}(𝔾), and (5.9) is preserved by sums, products and norm-limits. Fix for the remainder a = λ(v) as above, and put K = ∑_{l∈ℕ} α_{l}P̂_{l} ∈ B(L^{2}(𝔾)).

Since 1 ≤ \frac{m_{n}m_{k}}{m_{l}} = \frac{(1−q^{4n+2})(1−q^{4k+2})}{(1−q^{2})(1−q^{4l+2})} ≤ \frac{1}{(1−q^{2})^{2}} for k + n = l, it follows that (1 − q^{2})(l + 1)^{−1/2} ≤ α_{l} ≤ (l + 1)^{−1/2}. Thus K is compact with (unbounded) inverse K^{−1} = ∑_{l∈ℕ} α_{l}^{−1}P̂_{l}. Using K, we can write

aV^{∗} − V^{∗}(a ⊗ 1) = K(K^{−1}a − aK^{−1})V^{∗} − K(aK^{−1}V^{∗} − K^{−1}V^{∗}(a ⊗ 1)),

so it suffices to show that A_{1} = K^{−1}a−aK^{−1} and A_{2} = aK^{−1}V^{∗} −K^{−1}V^{∗}(a⊗ 1) are bounded linear maps. First consider A_{1}. Since a = λ(v) = P̂_{0}aP̂_{1} + ∑_{l≥1} ∑_{l′∈ℕ: |l−l′|≤1} P̂_{l}aP̂_{l′}, we have

A_{1} = (1 − α_{1}^{−1})P̂_{0}aP̂_{1} + ∑_{l≥1} ∑_{l′∈ℕ: |l−l′|≤1}(α_{l}^{−1} − α_{l′}^{−1})P̂_{l}aP̂_{l′}.

Quantum Automorphism Groups 1391

It therefore suffices to show that S := sup_{l≥1}{max_{|l′−l|≤1} |α_{l}^{−1} − α_{l′}^{−1}|} < ∞. Since (l + 1)^{1/2} ≤ α_{l}^{−1} ≤ \frac{(l+1)^{1/2}}{1−q^{2}} , the fact that S < ∞ is clear. Now consider A_{2}. Let â = F(v) ∈ B(H_{1}) ⊂ ℂ[𝔾̂], let y ∈ ℂ[𝔾̂] ⊗_{alg} ℂ[𝔾̂], and put y_{k,n} = (P̂_{k} ⊗ P̂_{n})y for each k, n ∈ ℕ. For each inclusion U^{l} ⊂ U^{k} ⊠ U^{n} (k, l, n ≥ 0), let ϕ_{l}^{k⊠n} = C_{(k,n,l)}^{−1/2}ρ_{l}^{k⊠n} : H_{l} → H_{k} ⊗ H_{n} be the (unique, up to multiplication by 𝕋) isometric morphism defined by (3.2)- (3.3). Since V^{∗}(y_{k,n}) = KP̂_{k+n}Conv(y_{k,n}) and (by Lemma 4.12 and its proof) P̂_{l}Conv(y_{k,n}) = \frac{m_{k}m_{n}}{m_{l}} (ϕ_{l}^{k⊠n})^{∗}y_{k,n}ϕ_{l}^{k⊠n} = \frac{m_{k}m_{n}}{m_{l}} Ad((ϕ_{l}^{k⊠n})^{∗})y_{k,n}, it follows that A_{2}(y_{k,n}) = Conv(â ⊗ P̂_{k+n}Conv(y_{k,n})) − K^{−1}V^{∗}((Conv ⊗ id)(â ⊗ y_{k,n})

= ∑_{r=0,±1} (P̂_{n+k+r}

× (Conv(â ⊗ P̂_{k+n}Conv(y_{k,n})) − Conv((P̂_{k+r}Conv ⊗ id)(â ⊗ y_{k,n}))))

= ∑_{r=0,±1} (\frac{m_{1}m_{k}m_{n}}{m_{n+k+r}} (Ad([(id ⊗ ϕ_{k+n}^{k⊠n})ϕ_{k+n+r}^{1⊠k+n}]^{∗})

− Ad([(ϕ_{k+r}^{1⊠k} ⊗ id)ϕ_{k+n+r}^{k+r⊠n}]^{∗}))[â ⊗ y_{k,n}]),

where P̂_{−1} is zero by convention. Let ϕ_{k+n+r,L}^{1⊠k⊠n} = (id ⊗ ϕ_{k+n}^{k⊠n})ϕ_{k+n+r}^{1⊠k+n} and ϕ_{k+n+r,R}^{1⊠k⊠n} = (ϕ_{k+r}^{1⊠k} ⊗ id)ϕ_{k+n+r}^{k+r⊠n}. By appealing to [36], Lemma A.1, note that we can find a constant C = C(q) > 0 (independent of k and n) such that

inf_{λ∈𝕋} ∥ϕ_{k+n+r,L}^{1⊠k⊠n} − λϕ_{k+n+r,R}^{1⊠k⊠n}∥_{TL_{2(k+n+r),2(k+n+1)}(δ)} ≤ Cq^{2k+r−1},  (5.10)

for each n, k ∈ ℕ and r = 0, ±1. Since we also have ∥T^{∗}(â ⊗ y_{k,n})S∥_{ℓ^{2}} ≤ (\frac{m_{n+k+r}}{m_{1}m_{k}m_{n}} )^{1/2}∥T ∥∥S∥∥â⊗y_{k,n}∥_{ℓ^{2}}^{⊗3} for any S, T ∈ Mor(H_{n+k+r}, H_{1}⊗H_{k}⊗H_{n}), an application of the triangle inequality and (5.10) yields

∥Ad([ϕ_{k+n+r,L}^{1⊠k⊠n}]^{∗}) − Ad([ϕ_{k+n+r,R}^{1⊠k⊠n}]^{∗})[â ⊗ y_{k,n}]∥_{ℓ^{2}(𝔾̂)}

≤ 2Cq^{2k+r−1}(\frac{m_{n+k+r}}{m_{1}m_{k}m_{n}} )^{1/2}∥â ⊗ y_{k,n}∥_{ℓ^{2}(𝔾̂)^{⊗3}}.

Using this last inequality and the Cauchy-Schwarz inequality, we obtain the

Documenta Mathematica 18 (2013) 1349–1402

estimate

∥A_{2}y∥_{ℓ^{2}(𝔾̂)}^{2} = ∑_{l≥0} ∥ ∑_{k,n:|n+k−l|≤1} P̂_{l}A_{2}(y_{k,n})∥_{ℓ^{2}(𝔾̂)}^{2}

= ∑_{l≥0} ∥ ∑_{k,n:|n+k−l|≤1} \frac{m_{1}m_{k}m_{n}}{m_{l}} (Ad([ϕ_{l,L}^{1⊠k⊠n}]^{∗}) − Ad([ϕ_{l,R}^{1⊠k⊠n}]^{∗}))[â ⊗ y_{k,n}]∥_{ℓ^{2}}^{2}

≤ ∑_{l≥0} ( ∑_{k,n:|n+k−l|≤1} 2Cq^{k+l−n−1}(\frac{m_{1}m_{k}m_{n}}{m_{l}} )^{1/2}∥â∥_{ℓ^{2}(𝔾̂)}∥y_{k,n}∥_{ℓ^{2}(𝔾̂)^{⊗2}})^{2}

≤ 4C^{2}m_{1}∥â∥_{ℓ^{2}(𝔾̂)}^{2}

× ∑_{l≥0} ( ∑_{k,n:|n+k−l|≤1} \frac{m_{k}m_{n}}{m_{l}} q^{2k+2l−2n−2})( ∑_{k,n:|k+n−l|≤1} ∥y_{n,k}∥_{ℓ^{2}(𝔾̂)^{⊗2}}^{2})

≤ 4C^{2}m_{1}∥â∥_{ℓ^{2}(𝔾̂)}^{2}

× sup_{l≥0} \{ ∑_{k,n:|n+k−l|≤1} \frac{m_{k}m_{n}}{m_{l}} q^{2k+2l−2n−2}\} ∑_{l≥0} ∑_{k,n:|k+n−l|≤1} ∥y_{n,k}∥_{ℓ^{2}(𝔾̂)^{⊗2}}^{2}.

Since the above sum over l is dominated by 3∥y∥_{ℓ^{2}(𝔾̂)^{⊗2}}^{2}, \frac{m_{k}m_{n}}{m_{l}} ≤ \frac{q^{−2}}{(1−q^{2})^{2}} for |k + n − l| ≤ 1, and

∑_{k,n:|n+k−l|≤1} \frac{m_{k}m_{n}}{m_{l}} q^{2k+2l−2n−2} ≤ ∑_{k,n:|n+k−l|≤1} \frac{q^{−2}}{(1 − q^{2})^{2}} q^{2k+2l−2n−2}

= \frac{q^{−4}}{(1 − q^{2})^{2}} ∑_{r=0,±1} ∑_{k=0}^{l+r} q^{4k−2r} ≤ \frac{3q^{−6}}{(1 − q^{2})^{2}(1 − q^{4})} (l ≥ 0),

we conclude that A_{2} is bounded. The proof of (5.9) when a = 1 and b ≠ 1 is essentially the same as the above argument. The only significant difference is that one must use a right-sided version of (5.10). The required inequality is given by Lemma A.2 from [36]. We leave the details to the reader.

5.5 Concluding Remark

Since this work was submitted for publication, the author has obtained a description of the quantum groups 𝔾_{aut}(B, ψ) for arbitrary invariant states ψ in terms of a free product ∗_{i}𝔾_{aut}(B_{i}, ψ_{i}), where each B_{i} is a certain distinguished subalgebra of B and ψ_{i} = ψ|_{B_{i}}. See Proposition 18 in [19].

6 Appendix – Proof of Theorem 5.4

In this section we prove Theorem 5.4. We will freely use the notation and assumptions of Section 5.1.

Proof of Theorem 5.4. We can write the commutator map T : L^{2}(𝔾) → H_{1} ⊗ L^{2}(𝔾)⊗H_{1} as the sum T = T_{L}−T_{R}, where T_{L} = ∑_{i,j=1}^{d_{1}} F_{1}e_{j} ⊗λ(u_{ij}^{1})⊗e_{i} is the part of T corresponding to the left action of {λ(u_{ij}^{1})}_{i,j=1}^{d_{1}} on L^{2}(𝔾) and −T_{R} is the remaining part of T corresponding to the right action of {λ(u_{ij}^{1})}_{i,j=1}^{d_{1}} on L^{2}(𝔾). Observe that T ξ_{0} = T_{L}ξ_{0} − T_{R}ξ_{0} = 0. In terms of matrix elements of irreducible representations of 𝔾, the map T_{L} (respectively T_{R}) corresponds to the operation of tensoring on the left (respectively on the right) by U^{1}. Since U^{1}⊠U^{k} ≅ U^{k+1}⊕U^{k}⊕U^{k−1} ≅ U^{k}⊠U^{1} for all k ≥ 1, we can uniquely decompose T_{L} and T_{R} as the sums T_{L,R} = ∑_{α=0,±1} T_{L,R}^{(α)}, where

T_{L,R}^{(0)}ξ_{0} = T_{L,R}^{(−1)}ξ_{0} = 0, T_{L}^{(+1)}ξ_{0} = T_{R}^{(+1)}ξ_{0} = ∑_{i,j=1}^{d_{1}} F_{1}e_{j} ⊗ λ(u_{ij}^{1})ξ_{0} ⊗ e_{i},

and for k ≥ 1, T_{L,R}^{(α)} : L_{k}^{2}(𝔾) → H_{1} ⊗ L_{k+α}^{2}(𝔾) ⊗ H_{1} is defined by

T_{L,R}^{(α)}|_{L_{k}^{2}(𝔾)} = (id_{H_{1}} ⊗ P_{k+α} ⊗ id_{H_{1}})T_{L,R}|_{L_{k}^{2}(𝔾)}.

Setting T^{(α)} := T_{L}^{(α)} − T_{R}^{(α)}, we obtain the decomposition T = ∑_{α=0,±1} T^{(α)} stated in Theorem 5.4. To verify the expressions for T^{(α)} given by Theorem 5.4, fix k ≥ 1, ξ, η ∈ H_{k} and consider the matrix element λ((ω_{η,ξ} ⊗ id)U^{k})ξ_{0} = Λ_{h}((ω_{η,ξ} ⊗ id)U^{k}) ∈ L_{k}^{2}(𝔾). In the following calculations, we freely use the identities given in Remarks 3.6 and 5.5, the unitary identifications

L_{k}^{2}(𝔾) ≅ H_{k}⊗H_{k}; λ((ω_{η,ξ}⊗id)U^{k})ξ_{0} ↦ ξ⊗(1⊗η^{∗})t_{2k} (ξ, η ∈ H_{k}, k ≥ 0),

given by (2.4), as well as the identities

∑_{j=1}^{d_{1}} F_{1}e_{j} ⊗ e_{j} = ∑_{j=1}^{d_{1}} F_{1}F_{1}^{∗}F_{1}^{tr}e_{j} ⊗ e_{j} = ∑_{j=1}^{d_{1}} F_{1}^{tr}e_{j} ⊗ e_{j} = [3]_{q}^{1/2}t_{2},

(p_{x} ⊗ p_{y})p_{x+y} = p_{x+y} = p_{x+y}(p_{x} ⊗ p_{y}) (x, y ∈ ℕ).

Since ⨁_{α=0,±1} ϕ_{k+α,L}^{(−α)} : ⨁_{α=0,±1} H_{k+α} → H_{1} ⊗ H_{k} is a unitary intertwiner, it follows that

T_{L}(λ((ω_{η,ξ} ⊗ id)U^{k})ξ_{0}) = ∑_{i,j=1}^{d_{1}} F_{1}e_{j} ⊗ λ(u_{ij}^{1}(ω_{η,ξ} ⊗ id)U^{k})ξ_{0} ⊗ e_{i}

= ∑_{i,j=1}^{d_{1}} F_{1}e_{j} ⊗ λ((ω_{e_{i}⊗η,e_{j}⊗ξ} ⊗ id)(U^{1} ⊠ U^{k}))ξ_{0} ⊗ e_{i}

= ∑_{α=0,±1} \underbrace{∑_{i,j=1}^{d_{1}} F_{1}e_{j} ⊗ λ((ω_{(ϕ_{k+α,L}^{(−α)})^{∗}(e_{i}⊗η),(ϕ_{k+α,L}^{(−α)})^{∗}(e_{j}⊗ξ)} ⊗ id)U^{k+α})ξ_{0} ⊗ e_{i}}_{∈H_{1}⊗L_{k+α}^{2}(𝔾)⊗H_{1}} .

1394 M. Brannan Therefore

T_{L}^{(α)}(ξ ⊗ (1 ⊗ η^{∗})t_{2k})

= ∑_{i,j=1}^{d_{1}} F_{1}e_{j} ⊗ λ((ω_{(ϕ_{k+α,L}^{(−α)})^{∗}(e_{i}⊗η),(ϕ_{k+α,L}^{(−α)})^{∗}(e_{j}⊗ξ)} ⊗ id)U^{k+α})ξ_{0} ⊗ e_{i}

= ∑_{j=1}^{d_{1}} F_{1}e_{j} ⊗ (ϕ_{k+α,L}^{(−α)})^{∗}(e_{j} ⊗ ξ) ⊗ ∑_{i=1}^{d_{1}}(1 ⊗ (e_{i}^{∗} ⊗ η^{∗})ϕ_{k+α,L}^{(−α)})t_{2k+2α} ⊗ e_{i}

= \underbrace{[(1_{2} ⊗ (ϕ_{k+α,L}^{(−α)})^{∗})( ∑_{j} F_{1}e_{j} ⊗ e_{j} ⊗ ξ)]}_{=:A_{1}^{(α)}ξ}

⊗ \underbrace{[(1_{2k+2α} ⊗ 1_{2} ⊗ η^{∗})(1_{2k+2α} ⊗ ϕ_{k+α,L}^{(−α)}))t_{2k+2α}]}_{=:A_{2}^{(α)}(1⊗η^{∗})t_{2k}},

and similarly

T_{R}^{(α)}(ξ ⊗ (1 ⊗ η^{∗})t_{2k}) = \underbrace{[(1_{2} ⊗ (ϕ_{k+α,R}^{(−α)})^{∗})( ∑_{j=1}^{d_{1}} F_{1}e_{j} ⊗ ξ ⊗ e_{j})]}_{=:B_{1}^{(α)}ξ}

⊗ \underbrace{[(1_{2k+2α} ⊗ η^{∗} ⊗ 1_{2})(1_{2k+2α} ⊗ ϕ_{k+α,R}^{(−α)})t_{2k+2α}]}_{=:B_{2}^{(α)}(1⊗η^{∗})t_{2k}} .

Let us first consider T_{L}^{(α)}|_{H_{k}⊗H_{k}} = A_{1}^{(α)} ⊗ A_{2}^{(α)}. When α = 1, we have

A_{1}^{(1)}ξ = [3]_{q}^{1/2}(1_{2} ⊗ (ϕ_{k+1,L}^{(−1)})^{∗})(t_{2} ⊗ ξ) = [3]_{q}^{1/2}(1_{2} ⊗ p_{2k+2}(p_{2} ⊗ p_{2k}))(t_{2} ⊗ ξ)

= [3]_{q}^{1/2}(p_{2} ⊗ p_{2k+2})(t_{2} ⊗ 1_{2k})p_{2k}ξ = (\frac{[2k + 3]_{q}}{[2k + 1]_{q}} )^{1/2}ϕ_{k,L}^{(+1)}ξ,

and A_{2}^{(1)}(1 ⊗ η^{∗})t_{2k} = (1_{2k+2} ⊗ 1_{2} ⊗ η^{∗})(1_{2k+2} ⊗ p_{2} ⊗ p_{2k})t_{2k+2}

= (\frac{[2k + 1]_{q}[3]_{q}}{[2k + 3]_{q}} )^{1/2}(1_{2k+2} ⊗ 1_{2} ⊗ η^{∗})(p_{2k+2} ⊗ p_{2} ⊗ p_{2k})(1_{2k} ⊗ t_{2} ⊗ 1_{2k})t_{2k}

= (1_{2k+2} ⊗ 1_{2} ⊗ η^{∗})(1_{2k+2} ⊗ 1_{2k+2})(ϕ_{k,R}^{(+1)} ⊗ 1_{2k})t_{2k} = ϕ_{k,R}^{(+1)}(1_{2k} ⊗ η^{∗})t_{2k}.

Documenta Mathematica 18 (2013) 1349–1402

When α = 0, we have

A_{1}^{(0)}ξ = [3]_{q}^{1/2}(p_{2} ⊗ (ϕ_{k,L}^{(0)})^{∗})(t_{2} ⊗ ξ)

= (\frac{[3]_{q}[2k]_{q}}{[2k + 2]_{q}} )^{1/2}(p_{2} ⊗ p_{2k}(m ⊗ 1_{2k−2})(p_{2} ⊗ p_{2k}))(t_{2} ⊗ ξ)

= (\frac{[3]_{q}[2k]_{q}}{[2k + 2]_{q}} )^{1/2}(p_{2} ⊗ p_{2k}(m ⊗ 1_{2k−2}))(t_{2} ⊗ ξ)

= (\frac{[2k]_{q}}{[2k + 2]_{q}} )^{1/2}(p_{2} ⊗ p_{2k}(m ⊗ 1_{2k−2}))(m^{∗}ν ⊗ 1_{2k})ξ

= (\frac{[2k]_{q}}{[2k + 2]_{q}} )^{1/2}(p_{2} ⊗ p_{2k})(1_{2} ⊗ m ⊗ 1_{2k−2})(m^{∗} ⊗ 1_{2k})(ν ⊗ 1_{2k})ξ

= (\frac{[2k]_{q}}{[2k + 2]_{q}} )^{1/2}(p_{2} ⊗ p_{2k})(m^{∗}m ⊗ 1_{2k−2})(ν ⊗ 1_{2k})ξ (cf. Remark 3.6)

= (\frac{[2k]_{q}}{[2k + 2]_{q}} )^{1/2}(p_{2} ⊗ p_{2k})(m^{∗} ⊗ 1_{2k−2})ξ = ϕ_{k,L}^{(0)}ξ,

and

A_{2}^{(0)}(1 ⊗ η^{∗})t_{2k} = (1_{2k} ⊗ 1_{2} ⊗ η^{∗})(1_{2k} ⊗ ϕ_{k,L}^{(0)})t_{2k} = ϕ_{k,R}^{(0)}(1_{2k} ⊗ η^{∗})t_{2k}.

The last equation follows because both of the isometries (ϕ_{k,R}^{(0)} ⊗ 1_{2k})t_{2k} and (1_{2k} ⊗ ϕ_{k,L}^{(0)})t_{2k} belong to the one-dimensional space Mor(1, U^{k} ⊠ U^{1} ⊠ U^{k}) and therefore (1_{2k} ⊗ ϕ_{k,L}^{(0)})t_{2k} = z(ϕ_{k,R}^{(0)} ⊗ 1_{2k})t_{2k} for some z ∈ 𝕋. But then

z = t_{2k}^{∗}((ϕ_{k,R}^{(0)})^{∗} ⊗ 1_{2k})(1_{2k} ⊗ ϕ_{k,L}^{(0)})t_{2k}

= \frac{[2k]_{q}}{[2k + 2]_{q}} t_{2k}^{∗}(1_{2k−2} ⊗ m ⊗ 1_{2k})(1_{2k} ⊗ p_{2} ⊗ 1_{2k})(1_{2k} ⊗ m^{∗} ⊗ 1_{2k−2})t_{2k}

= \frac{[2k]_{q}}{[2k + 2]_{q}} t_{2k}^{∗}(1_{2k−2} ⊗ m ⊗ 1_{2k})(1_{2k} ⊗ (1_{2} − νν^{∗}) ⊗ 1_{2k})

× (1_{2k} ⊗ m^{∗} ⊗ 1_{2k−2})t_{2k}

= \frac{[2k]_{q}}{[2k + 2]_{q}} (t_{2k}^{∗}(1_{2k−2} ⊗ m^{∗}m ⊗ 1_{2k−2})t_{2k} − t_{2k}^{∗}t_{2k})

= \frac{[2k]_{q}}{[2k + 2]_{q}} (\frac{[2]_{q}[2k + 1]_{q}}{[2k]_{q}} − 1) = 1,

where in the last line we have used

(1_{2k−2} ⊗ m ⊗ 1_{2k−2})t_{2k} = ([2]_{q}[2k + 1]_{q}[2k]_{q}^{−1})^{1/2}t_{2k−1}.

When α = −1, we have

A_{1}^{(−1)}ξ = [3]_{q}^{1/2}(1_{2} ⊗ (ϕ_{k−1,L}^{(+1)})^{∗})(t_{2} ⊗ ξ)

= (\frac{[3]_{q}^{2}[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}(1_{2} ⊗ p_{2k−2}(t_{2}^{∗} ⊗ 1_{2k−2})(p_{2} ⊗ p_{2k}))(t_{2} ⊗ ξ)

= (\frac{[3]_{q}^{2}[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}(p_{2} ⊗ p_{2k−2})(1_{2} ⊗ t_{2}^{∗} ⊗ 1_{2k−2})(t_{2} ⊗ 1_{2k})ξ

= (\frac{[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}(p_{2} ⊗ p_{2k−2})ξ = (\frac{[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}ϕ_{k,L}^{(−1)}ξ

(using (1_{2} ⊗ t_{2}^{∗})(t_{2} ⊗ 1_{2}) = [3]_{q}^{−1}p_{2}),

and

A_{2}^{(−1)}(1 ⊗ η^{∗})t_{2k}

= (\frac{[3]_{q}[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}

× (1_{2k−2} ⊗ 1_{2} ⊗ η^{∗})(p_{2k−2} ⊗ p_{2} ⊗ p_{2k})(1_{2k−2} ⊗ t_{2} ⊗ 1_{2k−2})t_{2k−2} = (1_{2k−2} ⊗ 1_{2} ⊗ η^{∗})(p_{2k−2} ⊗ p_{2} ⊗ 1_{2k})

× (\frac{[3]_{q}[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}(1_{2k} ⊗ p_{2k})(1_{2k−2} ⊗ t_{2} ⊗ 1_{2k−2})t_{2k−2}

= (1_{2k} ⊗ η^{∗})(ϕ_{k,R}^{(−1)} ⊗ 1_{2k})t_{2k} = ϕ_{k,R}^{(−1)}(1_{2k} ⊗ η^{∗})t_{2k}.

Let us now consider T_{R}^{(α)}|_{H_{k}⊗H_{k}} = B_{1}^{(α)} ⊗ B_{2}^{(α)}. When α = 1, we have

B_{1}^{(1)}ξ = [3]_{q}^{1/2}σ((ϕ_{k+1,R}^{(−1)})^{∗} ⊗ 1_{2})(ξ ⊗ t_{2}) = [3]_{q}^{1/2}σ(p_{2k+2}(p_{2k} ⊗ p_{2}) ⊗ 1_{2})(1_{2k} ⊗ t_{2})ξ = [3]_{q}^{1/2}σ(p_{2k+2} ⊗ p_{2})(1_{2k} ⊗ t_{2})ξ

= (\frac{[2k + 3]_{q}}{[2k + 1]_{q}} )^{1/2}σϕ_{k,R}^{(+1)}ξ,

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Quantum Automorphism Groups 1397

and B_{2}^{(1)}(1 ⊗ η^{∗})t_{2k} = (1_{2k+2} ⊗ η^{∗} ⊗ 1_{2})(1_{2k+2} ⊗ ϕ_{k+1,R}^{(−1)})t_{2k+2}

= (\frac{[2k + 1]_{q}}{[2k + 3]_{q}} )^{1/2}(1_{2k+2} ⊗ η^{∗} ⊗ 1_{2})(p_{2k+2} ⊗ 1_{2k+2})

× ( ∑_{j=1}^{d_{1}} e_{j} ⊗ 1_{2k} ⊗ 1_{2k} ⊗ F_{1}e_{j})t_{2k}

= (\frac{[2k + 1]_{q}}{[2k + 3]_{q}} )^{1/2}σ^{∗}(1_{2} ⊗ 1_{2k+2} ⊗ η^{∗})(1_{2} ⊗ p_{2k+2} ⊗ 1_{2k})

× ( ∑_{j=1}^{d_{1}} F_{1}e_{j} ⊗ e_{j} ⊗ 1_{2k} ⊗ 1_{2k})t_{2k}

= (\frac{[2k + 1]_{q}}{[2k + 3]_{q}} )^{1/2}σ^{∗}(1_{2} ⊗ p_{2k+2})( ∑_{j=1}^{d_{1}} F_{1}e_{j} ⊗ e_{j} ⊗ 1_{2k} ⊗ η^{∗})t_{2k}

= (\frac{[3]_{q}[2k + 1]_{q}}{[2k + 3]_{q}} )^{1/2}σ^{∗}(1_{2} ⊗ p_{2k+2})(t_{2} ⊗ 1_{2k} ⊗ η^{∗})t_{2k}

= (\frac{[3]_{q}[2k + 1]_{q}}{[2k + 3]_{q}} )^{1/2}σ^{∗}(1_{2} ⊗ p_{2k+2})(t_{2} ⊗ 1_{2k})(1_{2k} ⊗ η^{∗})t_{2k}

= σ^{∗}ϕ_{k,L}^{(+1)}((1_{2k} ⊗ η^{∗})t_{2k}). When α = 0, we have

B_{1}^{(0)}ξ = [3]_{q}^{1/2}σ((ϕ_{k,R}^{(0)})^{∗} ⊗ 1_{2})(ξ ⊗ t_{2})

= [3]_{q}^{1/2}(\frac{[2k]_{q}}{[2k + 2]_{q}} )^{1/2}σ(p_{2k}(1_{2k−2} ⊗ m)(p_{2k} ⊗ p_{2}) ⊗ 1_{2})(ξ ⊗ t_{2})

= [3]_{q}^{1/2}(\frac{[2k]_{q}}{[2k + 2]_{q}} )^{1/2}σ(p_{2k} ⊗ p_{2})(1_{2k−2} ⊗ m ⊗ 1_{2})(1_{2k} ⊗ t_{2})ξ

= (\frac{[2k]_{q}}{[2k + 2]_{q}} )^{1/2}σ(p_{2k} ⊗ p_{2})(1_{2k−2} ⊗ m ⊗ 1_{2})(1_{2k} ⊗ 1_{2} ⊗ p_{2})

× (1_{2k} ⊗ m^{∗})(1_{2k} ⊗ ν)ξ

= (\frac{[2k]_{q}}{[2k + 2]_{q}} )^{1/2}σ(p_{2k} ⊗ p_{2})(1_{2k−2} ⊗ m ⊗ 1_{2})(1_{2k} ⊗ m^{∗})(1_{2k} ⊗ ν)ξ

= (\frac{[2k]_{q}}{[2k + 2]_{q}} )^{1/2}σ(p_{2k} ⊗ p_{2})(1_{2k−2} ⊗ m^{∗})ξ = σϕ_{k,R}^{(0)}ξ.

Using the fact that

p_{2}^{⊗3}(1_{2} ⊗ m^{∗})t_{2} = [3]_{q}^{1/2}p_{2}^{⊗3}(1_{2} ⊗ m^{∗})m^{∗}ν
= [3]_{q}^{1/2}p_{2}^{⊗3}(m^{∗} ⊗ 1_{2})m^{∗}ν = p_{2}^{⊗3}(m^{∗} ⊗ 1_{2})t_{2},

Documenta Mathematica 18 (2013) 1349–1402

we also have

(\frac{[2k]_{q}}{[2k + 2]_{q}} )^{−1/2}B_{2}^{(0)}(1 ⊗ η^{∗})t_{2k}

= (1_{2k} ⊗ η^{∗} ⊗ 1_{2})(p_{2k} ⊗ (p_{2k} ⊗ p_{2})(1_{2k−2} ⊗ m^{∗}))t_{2k}

= (\frac{[3]_{q}[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}(p_{2k} ⊗ η^{∗} ⊗ p_{2})(1_{2k} ⊗ 1_{2k−2} ⊗ m^{∗})(1_{2} ⊗ t_{2k−2} ⊗ 1_{2})t_{2}

= (\frac{[3]_{q}[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}(p_{2k} ⊗ η^{∗} ⊗ p_{2})(1_{2} ⊗ t_{2k−2} ⊗ 1_{2} ⊗ 1_{2})(1_{2} ⊗ m^{∗})t_{2}

= (\frac{[3]_{q}[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}(p_{2k} ⊗ η^{∗} ⊗ p_{2})(1_{2} ⊗ t_{2k−2} ⊗ 1_{2} ⊗ 1_{2})(m^{∗} ⊗ 1_{2})t_{2}

= (\frac{[3]_{q}[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}σ^{∗}(p_{2} ⊗ p_{2k} ⊗ η^{∗})(1_{2} ⊗ 1_{2} ⊗ t_{2k−2} ⊗ 1_{2})(1_{2} ⊗ m^{∗})t_{2}

= (\frac{[3]_{q}[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}σ^{∗}(p_{2} ⊗ p_{2k} ⊗ η^{∗})(1_{2} ⊗ 1_{2} ⊗ t_{2k−2} ⊗ 1_{2})(m^{∗} ⊗ 1_{2})t_{2}

= (\frac{[3]_{q}[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}σ^{∗}(p_{2} ⊗ p_{2k} ⊗ η^{∗})(m^{∗} ⊗ 1_{4k−2})(1_{2} ⊗ t_{2k−2} ⊗ 1_{2})t_{2}

= σ^{∗}(p_{2} ⊗ p_{2k})(m^{∗} ⊗ 1_{2k−2})(1_{2k} ⊗ η^{∗})t_{2k}

= (\frac{[2k]_{q}}{[2k + 2]_{q}} )^{−1/2}σ^{∗}ϕ_{k,L}^{(0)}(1_{2k} ⊗ η^{∗})t_{2k}.

Finally, when α = −1, we have

B_{1}^{(−1)}ξ = [3]_{q}^{1/2}σ((ϕ_{k−1,R}^{(+1)})^{∗} ⊗ 1_{2})(ξ ⊗ t_{2})

= [3]_{q}^{1/2}(\frac{[3]_{q}[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}σ(p_{2k−2}(1_{2k−2} ⊗ t_{2}^{∗})(p_{2k} ⊗ p_{2}) ⊗ 1_{2})(ξ ⊗ t_{2})

= [3]_{q}^{1/2}(\frac{[3]_{q}[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}σ(p_{2k−2}(1_{2k−2} ⊗ t_{2}^{∗}) ⊗ 1_{2})(1_{2k} ⊗ t_{2})ξ

= [3]_{q}^{1/2}(\frac{[3]_{q}[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}σ(p_{2k−2} ⊗ 1_{2})(1_{2k−2} ⊗ t_{2}^{∗} ⊗ 1_{2})(1_{2k} ⊗ t_{2})ξ

= (\frac{[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}σ(p_{2k−2} ⊗ p_{2})ξ = (\frac{[2k − 1]_{q}}{[2k + 1]_{q}} )^{1/2}σϕ_{k,R}^{(−1)}ξ

(using [3]_{q}^{−1} = (t_{2}^{∗} ⊗ 1_{2})(1_{2} ⊗ t_{2})),

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and

(\frac{[3]_{q}[2k − 1]_{q}}{[2k + 1]_{q}} )^{−1/2}B_{2}^{(−1)}(1 ⊗ η^{∗})t_{2k}

= (1_{2k−2} ⊗ η^{∗} ⊗ 1_{2})(1_{2k−2} ⊗ (p_{2k} ⊗ p_{2})(1_{2k−2} ⊗ t_{2})p_{2k−2})t_{2k−2} = (p_{2k−2} ⊗ η^{∗} ⊗ p_{2})(1_{2k−2} ⊗ 1_{2k−2} ⊗ t_{2})t_{2k−2} = σ^{∗}(p_{2} ⊗ p_{2k−2} ⊗ η^{∗})(1_{2} ⊗ t_{2k−2} ⊗ 1_{2})t_{2}

= (\frac{[3]_{q}[2k − 1]_{q}}{[2k + 1]_{q}} )^{−1/2}σ^{∗}(p_{2} ⊗ p_{2k−2} ⊗ η^{∗})t_{2k}

= (\frac{[3]_{q}[2k − 1]_{q}}{[2k + 1]_{q}} )^{−1/2}σ^{∗}ϕ_{k,L}^{(−1)}(1_{2k} ⊗ η^{∗})t_{2k}.

The theorem now follows from these identities.

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Michael Brannan Department of Mathematics University of Illinois Urbana, IL 61801 USA mbrannan@illinois.edu