Twisted C^{∗}-Algebras Associated

to Finitely Aligned Higher-Rank Graphs

Aidan Sims, Benjamin Whitehead, and Michael F. Whittaker

Received: November 20, 2013 Revised: July 21, 2014

Communicated by Joachim Cuntz

Abstract. We introduce twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs and give a comprehensive treatment of their fundamental structural properties. We establish versions of the usual uniqueness theorems and the classification of gauge-invariant ideals. We show that all twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs are nuclear and satisfy the UCT, and that for twists that lift to real-valued cocycles, the K-theory of a twisted relative Cuntz-Krieger algebra is independent of the twist. In the final section, we identify a sufficient condition for simplicity of twisted Cuntz-Krieger algebras associated to higher-rank graphs which are not aperiodic. Our results indicate that this question is significantly more complicated than in the untwisted setting.

2010 Mathematics Subject Classification: 46L05 Keywords and Phrases: C^{∗}-algebra; graph algebra; Cuntz-Krieger algebra

This research was supported by the Australian Research Council.

1 Introduction

Twisted C^{∗}-Algebras of Finitely Aligned k-Graphs 833

In [12], Kumjian and Pask introduced higher-rank graphs (or k-graphs) and their C^{∗}-algebras. They considered only higher-rank graphs which are rowfinite and have no sources, the same simplifying assumptions that were made in the first papers on graph C^{∗}-algebras [14, 13, 1]. The theory was later expanded [23] to include the more general finitely aligned k-graphs; in this case the C^{∗}-algebraic relations that describe the Cuntz-Krieger algebra were determined by first analysing an associated Toeplitz algebra [22]. Interpolating between the Toeplitz algebra and the Cuntz-Krieger algebra are the relative Cuntz-Krieger algebras, which were introduced in [26], and then used in [25] to determine the gauge-invariant-ideal structure of the Cuntz-Krieger algebras of finitely aligned k-graphs. Simplicity of the Cuntz-Krieger algebra of a finitely aligned higher-rank graph was completely characterised in [18]. In [16], Kumjian, Pask and Sims studied the structure theory of the twisted Cuntz-Krieger algebra C^{∗}(Λ, c) associated to a row-finite higher-rank graph Λ with no sources and a 𝕋-valued categorical 2-cocycle c on Λ. They subsequently proved [17] that for cocycles c that lift to ℝ-valued cocycles, the K-theory of C^{∗}(Λ, c) is the same as that of C^{∗}(Λ). In this paper, we extend these results to finitely aligned k-graphs, and identify the gauge-invariant ideal structure of twisted relative Cuntz-Krieger algebras of higher-rank graphs (the Cuntz-Krieger algebra and the Toeplitz algebra are special cases). We also establish a sufficient condition for simplicity of C^{∗}(Λ, c) when Λ is row-finite with no sources and c is induced by the degree map from a cocycle on ℤ^{k}. The sufficient condition for simplicity is new and requires an intricate analysis of the subalgebra generated by spanning elements in C^{∗}(Λ, c) whose initial and final projections coincide, and relies on a local decomposition of this subalgebra as a direct sum of noncommutative tori. We have organised this paper as follows. Section 2 introduces the necessary background about k-graphs and their cohomology. In Section 3, we introduce the twisted Toeplitz algebra, the twisted relative Cuntz-Krieger algebras and the twisted Cuntz-Krieger algebra of a finitely aligned k-graph with respect to a 𝕋-valued cocycle c. Following the program of [22, 25] we prove a version of Coburn’s theorem for the twisted Toeplitz algebra, and versions of an Huef and Raeburn’s gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem for the twisted relative Cuntz-Krieger algebras. We give a sufficient condition for the twisted Cuntz-Krieger algebra to be simple and purely infinite. In Section 4, we adapt the analysis of [25] to give a complete listing of the gauge-invariant ideals in a twisted relative Cuntz-Krieger algebra. This is new even in the untwisted setting, since the results of [25] only apply to the Cuntz-Krieger algebra. In Section 5, we modify arguments of [25, §8] to show that every twisted relative Cuntz-Krieger algebra is Morita equivalent to a crossed product of an AF algebra by ℤ^{k}, and is therefore nuclear and satisfies the UCT. In Section 6, we combine ideas from [23] and [17] to show that if the twisting cocycle arises by exponentiation of a real-valued 2-cocycle, then the K-groups of the twisted relative Cuntz-Krieger algebra are isomorphic to those of the corresponding untwisted algebra. Since our proofs of the results discussed in the preceding paragraph follow the lines of proofs of earlier results, our treatment is mostly quite brief, relying heavily on reference to the existing arguments, and we provide additional detail only where a nontrivial change is needed. One (perhaps surprising) example of the latter is the matter of ascertaining which diagonal projections in a twisted relative Cuntz-Krieger algebra are nonzero; it turns out that neither the arguments used for the row-finite case in [16] nor those used for the untwisted case in [26] can easily be adapted to our setting, so we use a different approach employing filters (see [9] and [3]) in a k-graph Λ and the path-space representation. This approach simplifies and streamlines even the untwisted setting [26], and substantially improves upon the argument used for twisted C^{∗}-algebras of row-finite k-graphs with no sources in [16]. In the final section, we consider simplicity of twisted k-graph C^{∗}-algebras. For untwisted C^{∗}-algebras it is proved in [18] that the Cuntz-Krieger algebra of a finitely aligned k-graph Λ is simple if and only if Λ is cofinal and aperiodic (other relative Cuntz-Krieger algebras are never simple). In the twisted situation, the “if” implication in this result follows from more or less the same argument (see Corollary 3.18); and necessity of cofinality also persists, although the argument of [18] requires some modification. However, for twisted C^{∗}-algebras, aperiodicity of Λ is not necessary for simplicity of C^{∗}(Λ, c): when ℕ^{2} is regarded as a 2-graph it is certainly not aperiodic, but its twisted C^{∗}-algebras are the rotation algebras (see [15, Example 7.7]), whose simplicity or otherwise depends on the twisting cocycle [27]. Recent work on primitive ideals in k-graph C^{∗}- algebras [4] shows that if Λ is row-finite with no sources and is cofinal, then the primitive ideals of C^{∗}(Λ) are indexed by characters of a subgroup Per(Λ) of ℤ^{k}. We consider cofinal row-finite k-graphs with no sources, and cocycles which are pulled back along the degree map from cocycles c of ℤ^{k}. We show that if the associated skew-symmetric bicharacter cc^{∗} restricts to a nondegenerate bicharacter of Per(Λ) (see [19, 20, 27]) — the condition which characterises simplicity of the noncommutative torus C^{∗}(Per(Λ), c) — then the associated twisted k-graph C^{∗}-algebra is simple.

We thank the anonymous referee for detailed and helpful comments; in particular the referee’s suggestions have substantially improved the presentation of the results in Section 7.

2 Background

Throughout this paper, ℕ denotes the natural numbers including 0, and ℕ^{k} is the monoid of k-tuples of natural numbers under coordinatewise addition. We denote the generators of ℕ^{k} by e_{1}, …, e_{k} and we write n_{i} for the i^{th} coordinate of n ∈ ℕ^{k}, so that n = ∑_{i} n_{i} · e_{i}. For m, n ∈ ℕ^{k}, we write m ∨ n and m ∧ n for their coordinatewise maximum and minimum. We regard ℕ^{k} as a partially ordered set with m ≤ n if and only if m_{i} ≤ n_{i} for all i. When convenient we will regard ℕ^{k} as a category with one object and composition given by addition. A k-graph is a countable category Λ endowed with a functor d : Λ → ℕ^{k} that satisfies the following factorisation property: whenever d(λ) = m + n there are unique μ ∈ d^{−1}(m) and ν ∈ d^{−1}(n) such that λ = μν. We write Λ^{n} for d^{−1}(n). Since the identity morphisms of Λ are idempotents and d is a functor, the identity morphisms belong to Λ^{0}. Hence the codomain and domain maps in the category Λ determine maps r, s : Λ → Λ^{0}. We then have r(λ)λ = λ = λs(λ) for all λ, and the factorisation property implies that Λ^{0} = {id_{o} : o ∈ Obj(Λ)}.

Since we are thinking of Λ as a kind of graph, we call its morphisms paths; the paths Λ^{0} of degree 0 are called vertices. Given λ ∈ Λ and E ⊆ Λ we write λE := {λμ : μ ∈ E, r(μ) = s(λ)}, and Eλ is defined similarly. In particular, if v ∈ Λ^{0} and n ∈ ℕ^{k} then vΛ^{n} = {λ ∈ Λ^{n} : r(λ) = v}. For μ, ν ∈ Λ, we define

MCE(μ, ν) := {λ ∈ Λ^{d(μ)∨d(ν)} : λ = μα = νβ for some α, β}
= μΛ ∩ νΛ ∩ Λ^{d(μ)∨d(ν)}.

We say that Λ is finitely aligned if MCE(μ, ν) is always finite (possibly empty). Let v ∈ Λ^{0} and E ⊆ vΛ. We say that E is exhaustive if for every λ ∈ vΛ there exists μ ∈ E such that MCE(λ, μ) ≠ ∅. Equivalently, E is exhaustive if EΛ ∩ λΛ ≠ ∅ for all λ ∈ vΛ. Following [26], we write FE(Λ) for the collection of all finite exhaustive sets E in Λ such that E ∩ Λ^{0} = ∅. Each E ∈ FE(Λ) is a subset of vΛ for some v. Given any v ∈ Λ^{0} and E ⊆ vΛ, we write r(E) := v. If E ∈ FE(Λ) with r(E) = v, we say E ∈ v FE(Λ). If E ⊆ FE(Λ) and v ∈ Λ^{0}, then vE = E ∩ v FE(Λ). Following [16], if A is an abelian group, we say that c : {(μ, ν) ∈ Λ × Λ : s(μ) = r(ν)} → A is a normalised cocycle if c(λ, μ) + c(λμ, ν) = c(μ, ν) + c(λ, μν) for all composable λ, μ, ν, and c(r(λ), λ) = 0 = c(λ, s(λ)) for all λ. (In the special case A = 𝕋, we will write the group operation multiplicatively and the identity element as 1, and write \overline{c} for the inverse \overline{c}(λ, μ) = \overline{c(λ, μ)} of c ∈ \underline{Z}^{2}(Λ, 𝕋).) We write \underline{Z}^{2}(Λ, A) for the collection of all normalised cocycles, which forms a group under pointwise addition in A. In the cohomology introduced in [16], the coboundary map \underline{δ}^{1} carries a function b : Λ → A to the map \underline{δ}^{1}b ∈ \underline{Z}^{2}(Λ, A) given by (\underline{δ}^{1}b)(μ, ν) = b(μ) + b(ν) − b(μν). The elements of the range of \underline{δ}^{1} are called coboundaries. Cocycles c, c′ ∈ \underline{Z}^{2}(Λ, A) are cohomologous if c′ − c is a coboundary.

3 The core and the uniqueness theorems

In this section we prove uniqueness theorems for twisted relative k-graph C^{∗}- algebras: a version of Coburn’s theorem for the twisted Toeplitz algebra, and versions of the gauge-invariant uniqueness theorem and Cuntz-Krieger uniqueness theorem for twisted relative Cuntz-Krieger algebras. We finish with a sufficient condition for the twisted Cuntz-Krieger algebra of a k-graph to be simple and purely infinite. The following definition parallels [22, Definition 7.1].

Definition 3.1. Let Λ be a finitely aligned k-graph and let c ∈ \underline{Z}^{2}(Λ, 𝕋). A Toeplitz-Cuntz-Krieger (Λ, c)-family in a C^{∗}-algebra A is a collection t = {t_{λ} : λ ∈ Λ} ⊆ A satisfying

(TCK1) {t_{v} : v ∈ Λ^{0}} is a set of mutually orthogonal projections; (TCK2) t_{μ}t_{ν} = c(μ, ν)t_{μν} whenever s(μ) = r(ν);

(TCK3) t_{λ}^{∗}t_{λ} = t_{s(λ)} for all λ ∈ Λ; and

(TCK4) t_{μ}t_{μ}^{∗}t_{ν}t_{ν}^{∗} = ∑_{λ∈MCE(μ,ν)} t_{λ}t_{λ}^{∗} for all μ, ν ∈ Λ, where empty sums are interpreted as zero.

We write C^{∗}(t) for C^{∗}({t_{λ} : λ ∈ Λ}) ⊆ A.

By [21, Proposition A.4], relation (TCK3) ensures that the t_{λ} are partial isometries, so that t_{λ}t_{λ}^{∗}t_{λ} = t_{λ} for all λ.

Lemma 3.2. Let Λ be a finitely aligned k-graph, c ∈ \underline{Z}^{2}(Λ, 𝕋) and t a Toeplitz-Cuntz-Krieger (Λ, c)-family. The projections {t_{μ}t_{μ}^{∗} : μ ∈ Λ} pairwise commute, and {t_{μ}t_{μ}^{∗} : μ ∈ Λ^{n}} is a collection of mutually orthogonal projections for each n ∈ ℕ^{k}. For μ, ν, η, ζ ∈ Λ we have

t_{ν}^{∗}t_{η} = ∑_{να=ηβ∈MCE(ν,η)} \overline{c(ν, α)}c(η, β)t_{α}t_{β}^{∗} and
t_{μ}t_{ν}^{∗}t_{η}t_{ζ}^{∗} = ∑_{να=ηβ∈MCE(ν,η)} c(μ, α)\overline{c(ν, α)}c(η, β)\overline{c(ζ, β)}t_{μα}t_{ζβ}^{∗}.

In particular C^{∗}(t) = \overline{span}\{t_{μ}t_{ν}^{∗} : s(μ) = s(ν)\}.

Proof. The first assertion follows from (TCK4) as MCE(μ, ν) = MCE(ν, μ). If d(μ) = d(ν) and μ ≠ ν, then the factorisation property ensures that μΛ ∩ νΛ = ∅, and in particular MCE(μ, ν) = ∅. Hence t_{μ}t_{μ}^{∗}t_{ν}t_{ν}^{∗} = 0 by (TCK4); so the t_{μ}t_{μ}^{∗} where μ ∈ Λ^{m} are mutually orthogonal. For the first displayed equation, we use (TCK4), (TCK2), and then (TCK2) to calculate:

t_{ν}^{∗}t_{η} = t_{ν}^{∗}(t_{ν}t_{ν}^{∗}t_{η}t_{η}^{∗})t_{η} = ∑_{να=ηβ∈MCE(ν,η)} t_{ν}^{∗}t_{να}t_{ηβ}^{∗}t_{η}

= ∑_{να=ηβ∈MCE(ν,η)} \overline{c(ν, α)}c(η, β)t_{ν}^{∗}t_{ν}t_{α}t_{β}^{∗}t_{η}^{∗}t_{η}

= ∑_{να=ηζ∈MCE(ν,η)} \overline{c(ν, α)}c(η, β)t_{α}t_{β}^{∗}.

Multiplying this expression on the left by t_{μ} and on the right by t_{ζ} and then applying (TCK2) yields the second displayed equation. It follows that \overline{span}\{t_{μ}t_{ν}^{∗} : μ, ν ∈ Λ\} is closed under multiplication and hence equal to C^{∗}(t), so the final assertion follows from the observation that if s(μ) ≠ s(ν), then t_{μ}t_{ν}^{∗} = t_{μ}t_{s(μ)}t_{s(ν)}t_{ν}^{∗} = 0 by (TCK3) and (TCK1).

As a consequence of Lemma 3.2, products of the form ∏_{λ∈E}(t_{v} −t_{λ}t_{λ}^{∗}) for finite subsets E of Λ are well-defined, and so we can make the following definition.

Definition 3.3. For E ⊆ FE(Λ), we say that a Toeplitz Cuntz-Krieger (Λ, c)- family t is a relative Cuntz-Krieger (Λ, c; E)-family if

(CK) for every v ∈ E^{0} and E ∈ vE, we have ∏_{λ∈E}(t_{r(E)} − t_{λ}t_{λ}^{∗}) = 0.

A relative Cuntz-Krieger (Λ, c; FE(Λ))-family is called a Cuntz-Krieger (Λ, c)- family.

Remark 3.4. If v ∈ Λ^{0} and r(λ) = v then we have t_{v}t_{λ}t_{λ}^{∗} = c(v, λ)t_{vλ}t_{λ}^{∗} = t_{λ}t_{λ}^{∗}. Lemma 3.2 implies in particular that {t_{λ}t_{λ}^{∗} : λ ∈ vΛ^{n}} is a set of mutually orthogonal projections, so ∑_{λ∈F} t_{λ}t_{λ}^{∗} is a projection for every finite F ⊆ vΛ^{n}.

Since t_{v}( ∑_{λ∈F} t_{λ}t_{λ}^{∗}) = ∑_{λ∈F} t_{λ}t_{λ}^{∗}, we conclude that t_{v} ≥ ∑_{λ∈F} t_{λ}t_{λ}^{∗}. So relations (TCK1)–(TCK4) (for c ≡ 1) imply relation (4) of [22, Definition 7.1], and in particular a Toeplitz-Cuntz-Krieger (Λ, 1)-family as defined here is the same thing as a Toeplitz-Cuntz-Krieger Λ-family in the sense of [22]. Lemma 3.2 also shows that a relative Cuntz-Krieger (Λ, 1; E)-family is the same thing as a relative Cuntz-Krieger Λ-family in the sense of [26]

The t_{λ} in any Toeplitz-Cuntz-Krieger (Λ, c)-family are partial isometries, and hence have norm 0 or 1, and the same is then true for the t_{μ}t_{ν}^{∗}. It is straightforward (following the strategy of, for example, [21, Propositions 1.20 and 1.21]) to show that there is a C^{∗}-algebra T C^{∗}(Λ, c) generated by a Toeplitz-Cuntz-Krieger family s_{T}^{c} := {s_{T}^{c} (λ) : λ ∈ Λ} which is universal in the sense that every Toeplitz-Cuntz-Krieger (Λ, c)-family t induces a homomorphism π_{t} : T C^{∗}(Λ, c) → C^{∗}(t) satisfying π_{t}(s_{T}^{c} (λ)) = t_{λ} for all λ. Given a set E ⊆ FE(Λ), let J_{E} ⊆ T C^{∗}(Λ, c) be the ideal generated by { ∏_{λ∈E}(s_{T}^{c} (r(E)) − s_{T}^{c} (λ)s_{T}^{c} (λ)^{∗}) : E ∈ E}. Then the quotient

C^{∗}(Λ, c; E) := T C^{∗}(Λ, c)/J_{E}

is, by construction, universal for relative Cuntz-Krieger (Λ, c; E)-families. We denote each s_{T}^{c} (λ) + J_{E} by s_{E}^{c}(λ), so that s_{E}^{c} := {s_{E}^{c}(λ) : λ ∈ Λ} is a universal (Λ, c; E)-family in C^{∗}(Λ, c; E). In the special case where E = FE(Λ) we simply write s^{c}(λ) for s_{FE(Λ)}^{c}(λ), and we denote C^{∗}(Λ, c; FE(Λ)) by C^{∗}(Λ, c). It follows almost immediately from the universal property that C^{∗}(Λ, c; E) depends only on the cohomology class of c:

Proposition 3.5. Let Λ be a finitely aligned k-graph and let E be a subset of FE(Λ). Suppose that c_{1}, c_{2} ∈ \underline{Z}^{2}(Λ, 𝕋) are cohomologous; say c_{1} = \underline{δ}^{1}bc_{2}. Then there is an isomorphism π : C^{∗}(Λ, c_{1}; E) → C^{∗}(Λ, c_{2}; E) such that π(s_{E}^{c_{1}}(λ)) = b(λ)s_{E}^{c_{2}}(λ) for each λ ∈ Λ.

Proof. The proof is essentially that of [16, Proposition 5.6]: the formula t_{λ} := b(λ)s_{E}^{c_{2}}(λ) defines a relative Cuntz-Krieger (Λ, c_{1}; E)-family in C^{∗}(Λ, c_{2}; E) and therefore induces a homomorphism π carrying each s_{E}^{c_{1}}(λ) to b(λ)s_{E}^{c_{2}}(λ). Interchanging the roles of c_{1} and c_{2} and replacing b with \overline{b} yields an inverse for π, and the result follows.

The universal property of each C^{∗}(Λ, c; E) ensures that there is a homomorphism γ^{c} : 𝕋^{k} → Aut(C^{∗}(Λ, c; E)) such that γ_{z}^{c}(s_{E}^{c}(λ)) = z^{d(λ)}s_{E}^{c}(λ) for all λ. An ε/3-argument shows that γ^{c} is strongly continuous. Averaging over γ^{c} gives a faithful conditional expectation

Φ_{E}^{c} : C^{∗}(Λ, c; E) → C^{∗}(Λ, c; E)^{γ^{c}} := {a ∈ C^{∗}(Λ, c; E) : γ_{z}^{c}(a) = a for all z}.

Since γ_{z}^{c}(s_{E}^{c}(μ)s_{E}^{c}(ν)^{∗}) = z^{d(μ)−d(ν)}s_{E}^{c}(μ)s_{E}^{c}(ν)^{∗} and since the s_{E}^{c}(μ)s_{E}^{c}(ν)^{∗} span a dense subspace of C^{∗}(Λ, c; E), we see that Φ_{E}^{c} is characterised by

Φ_{E}^{c}(s_{E}^{c}(μ)s_{E}^{c}(ν)^{∗}) = δ_{d(μ),d(ν)}s_{E}^{c}(μ)s_{E}^{c}(ν)^{∗},

and so C^{∗}(Λ, c; E)^{γ^{c}} = \overline{span}\{s_{E}^{c}(μ)s_{E}^{c}(ν)^{∗} : d(μ) = d(ν)\}. For E ⊆ r(λ)Λ, we write

Ext(λ; E) := ⋃_{μ∈E}{α : λα ∈ MCE(λ, μ)}.

We say that a subset E of FE(Λ) is satiated if it satisfies

(S1) if F ∈ E and λ ∈ r(F)Λ \ {r(F)}, then F ∪ {λ} ∈ E;

(S2) if F ∈ E and λ ∈ r(F)Λ \ F Λ, then Ext(λ; F) ∈ E;

(S3) if F ∈ E and λ, λλ′ ∈ F with λ′ ≠ s(λ), then F \ {λλ′} ∈ E;

(S4) if F ∈ E, λ ∈ F and G ∈ E with r(G) = s(λ), then F \ {λ} ∪ λG ∈ E.

Lemma 5.3 of [26] shows that the sets constructed in (S1)–(S4) belong to FE(Λ), and so FE(Λ) always contains E and satisfies (S1)–(S4). Given E ⊆ FE(Λ), we write \overline{E} for the intersection of all satiated subsets of FE(Λ) containing E. We call \overline{E} the satiation of E. Our (S1)–(S4) are different from those of [26]; but any set that can be constructed by our (S1)–(S4) can be constructed from an application of the corresponding operation from [26], and conversely any set that can be constructed by any of (S1)–(S4) from [26] can be obtained from finitely many applications of the operations discussed above. So our definition of a satiated set agrees with that of [26].

Notation 3.6. Let Λ be a finitely aligned k-graph, let c ∈ \underline{Z}^{2}(Λ, 𝕋) and let t be a Toeplitz-Cuntz-Krieger (Λ, c)-family. For λ ∈ Λ, we write q_{λ} := t_{λ}t_{λ}^{∗}. For v ∈ Λ^{0} and a finite subset E of vΛ, we define

Δ(t)^{E} := ∏_{λ∈E}(q_{r(E)} − q_{λ}).

In the universal algebras C^{∗}(Λ, c; E), we write p_{E}^{c}(λ) := s_{E}^{c}(λ)s_{E}^{c}(λ)^{∗} ∈ C^{∗}(Λ, c; E), and then Δ(s_{E}^{c})^{E} = ∏_{λ∈E} (p_{E}^{c}(r(E)) − p_{E}^{c}(λ)).

Lemma 3.7. Let Λ be a finitely aligned k-graph, let c ∈ \underline{Z}^{2}(Λ, 𝕋) and let t be a Toeplitz-Cuntz-Krieger (Λ, c)-family. If ∅ ≠ E ⊆ vΛ is finite and μ ∈ vΛ, then

Δ(t)^{E}t_{μ} = t_{μ}Δ(t)^{Ext(μ;E)}.

Proof. Fix λ ∈ E and use Lemma 3.2 to calculate

Δ(t)^{E}t_{μ} = Δ(t)^{E\{λ}}(q_{v} − t_{λ}t_{λ}^{∗})t_{μ}t_{s(μ)}^{∗}

= Δ(t)^{E\\{λ\}}t_{μ} − ∑_{λα=μβ∈MCE(λ,μ)} c(λ, α)\overline{c(λ, α)}c(μ, β)\overline{c(s(μ), β)}t_{λα}t_{β}^{∗}.

For each α, β we have t_{λα} = t_{μβ} = \overline{c(μ, β)}t_{μ}t_{β}, and we deduce that

Δ(t)^{E}t_{μ} = Δ(t)^{E\{λ}}t_{μ}(q_{s(μ)} − ∑_{β∈Ext(μ;{λ})} q_{β}).

The elements β ∈ Ext(μ; {λ}) all have degree (d(μ) ∨ d(λ)) − d(μ), and so the q_{β} are mutually orthogonal. Hence q_{s(μ)} − ∑_{β∈Ext(μ;{λ})} q_{β} = ∏_{β∈Ext(μ;{λ})}(q_{s(μ)} − q_{β}). Now an induction on |E| using that Ext(μ; E) = ⋃_{λ∈E} Ext(μ; {λ}) proves the lemma.

Proposition 3.12 of [10] implies that Ext(μν; E) = Ext(ν; Ext(μ; E)) for all composable μ, ν. If μ ∈ EΛ, then Δ(s_{T}^{c})^{E} ≤ s_{T}^{c} (r(μ)) − p_{T}^{c} (μ), forcing Δ(s_{T}^{c})^{E}s_{T}^{c} (μ) = 0. So Lemma 3.7 and condition (S2) imply that for a satiated subset E ⊆ FE(Λ), the ideal J_{E} generated by {Δ(s_{T}^{c})^{E} : E ∈ E} satisfies

J_{E} = \overline{span}\{s_{T}^{c} (μ)Δ(s_{T}^{c})^{E}s_{T}^{c} (ν)^{∗} : s(μ) = s(ν) and E ∈ s(μ)E\}.  (3.1)

We introduce the notion of a filter on a k-graph. For details, see [3, Section 3] (taking P = ℕ^{k}). We call a subset S of Λ a filter if

(F1) λΛ ∩ S ≠ ∅ implies λ ∈ S; and

(F2) μ, ν ∈ S implies MCE(μ, ν) ∩ S ≠ ∅.

For μ, ν ∈ Λ, we write μ ≤ ν if ν = μν′. Since λ ∈ MCE(μ, ν) implies μ, ν ≤ λ, Condition (F2) implies that each (S, ≤) is a directed set. Furthermore, for μ, ν ∈ Λ, distinct elements of MCE(μ, ν) have the same degree, and so themselves have no common extensions. Hence condition (F2) implies that | MCE(μ, ν) ∩ S| = 1 for μ, ν ∈ S. For a satiated set E ⊆ FE(Λ), we say that a filter S is \overline{E}-compatible if, whenever λ ∈ S and E ∈ s(λ)\overline{E}, we have λE ∩ S ≠ ∅.

Remark 3.8. It is straightforward to check that the \overline{E}-compatible boundary paths of [26] are in bijection with \overline{E}-compatible filters via the map x ↦ \{x(0, n) : n ≤ d(x)}.

Let {h_{λ} : λ ∈ Λ} be the usual orthonormal basis for ℓ^{2}(Λ). Routine calculations show that the formula T_{μ}h_{ν} := δ_{s(μ),r(ν)}c(μ, ν)h_{μν} determines a Toeplitz-Cuntz-Krieger (Λ, c)-family in B(ℓ^{2}(Λ)). Let π_{T} : T C^{∗}(Λ, c) → B(ℓ^{2}(Λ)) be the representation induced by the universal property of T C^{∗}(Λ, c).

Proposition 3.9. Let Λ be a finitely aligned k-graph and let c ∈ \underline{Z}^{2}(Λ, 𝕋). Let E ⊆ FE(Λ). Then the s_{E}^{c}(v) are all nonzero, and for each v ∈ Λ^{0} and finite F ⊆ vΛ, we have Δ(s_{E}^{c})^{F} = 0 if and only if either v ∈ F or F ∈ \overline{E}. We have J_{E} = J\overline{E} and C^{∗}(Λ, c; E) = C^{∗}(Λ, c; \overline{E}).

To prove the proposition, we need to be able to tell when a ∈ T C^{∗}(Λ, c) does not belong to J_{E}. We present the requisite statement as a separate result because we will use it again in Section 7 (see Lemma 7.2).

Proposition 3.10. Let Λ be a finitely aligned k-graph and let c ∈ \underline{Z}^{2}(Λ, 𝕋). Let E ⊆ FE(Λ). Then

J_{E} = \{a ∈ T C^{∗}(Λ, c) : lim_{λ∈S} π_{T} (a)h_{λ} = 0 for every \overline{E}-compatible filter S\}.

Here we only need the “⊆” containment — we will use it immediately to establish Proposition 3.9, which we will use in turn to prove the gauge-invariant uniqueness theorem and then our characterisation of the gauge-invariant ideals in C^{∗}(Λ, c; E). With this catalogue of gauge-invariant ideals in hand, it is then easy to establish the reverse inclusion. So we prove the “⊆” inclusion now, and defer the proof of the “⊇” inclusion until after Theorem 4.6.

Proof of ⊆ in Proposition 3.10. Fix v ∈ Λ^{0}, an element E ∈ E with r(E) = v and an \overline{E}-compatible filter S. We claim that Δ(T)^{E}h_{λ} = 0 for large λ ∈ S. To see this, we consider two cases. First suppose that v ∉ S. Then (F1) ensures that S ∩ vΛ = ∅ and so T_{v}h_{λ} = 0 for all λ ∈ S. Since Δ(T)^{E} ≤ T_{v}, it follows that Δ(T)^{E}h_{λ} = 0 for all λ ∈ S. Now suppose that v ∈ S. Since S is \overline{E}- compatible and E ∈ E, there exists λ ∈ E ∩ S. Hence Δ(T)^{E} ≤ T_{v} − T_{λ}T_{λ}^{∗}. For μ ∈ S with λ ≤ μ, we therefore have Δ(T)^{E}h_{μ} = 0. Hence Δ(T)^{E}h_{λ} = 0 for large λ ∈ S, proving the claim. It is now elementary to deduce that any finite linear combination of the form a = ∑_{μ,ν∈F} a_{μ,ν}s_{T}^{c} (μ)Δ(s_{T}^{c})^{E}s_{T}^{c} (ν)^{∗} where the E belong to E satisfies π_{T} (a)h_{λ} = 0 for large λ ∈ E. A continuity argument (details appear in [28, Lemma 3.4.14]) using this and (3.1) then completes the proof.

Lemma 3.11. Let Λ be a finitely aligned k-graph and let c ∈ \underline{Z}^{2}(Λ, 𝕋). Let E ⊆ FE(Λ). For each v ∈ Λ^{0} there exists an \overline{E}-compatible filter S such that ∥T_{v}h_{λ}∥ = 1 for all λ ∈ S, and for each E ∈ FE(Λ) \ \overline{E}, there exists an \overline{E}- compatible filter S such that ∥Δ(T)^{E}h_{λ}∥ = 1 for all λ ∈ S.

Proof. Fix v ∈ Λ^{0}. The argument of [26, Lemma 4.7] shows that there exists an \overline{E}-compatible filter S that contains v. Hence ∥T_{v}h_{λ}∥ = ∥h_{λ}∥ = 1 for all λ ∈ S. Now fix E ∈ FE(Λ) \ E. Then the argument of [26, Lemma 4.7] implies that there is an \overline{E}-compatible filter S such that r(E) ∈ S but EΛ∩S = ∅. Thus T_{μ}T_{μ}^{∗}h_{λ} = 0 for all μ ∈ E and λ ∈ S, and so ∥Δ(T)^{E}h_{λ}∥ = ∥T_{r(E)}h_{λ}∥ = 1 for all λ ∈ S.

Proof of Proposition 3.9. For the “if” implication, observe that if v ∈ F then Δ_{v} ≤ q_{v} − q_{v} = 0. To see that F ∈ \overline{E} implies Δ(s_{E}^{c})^{F} = 0, one checks that the calculations in [26] that establish the corresponding statement for the untwisted C^{∗}-algebra C^{∗}(Λ; E) are also valid in the twisted C^{∗}-algebra. Since we clearly have J_{E} ⊆ J\overline{E} we deduce from this that J_{E} = J\overline{E}, and therefore that C^{∗}(Λ, c; E) = C^{∗}(Λ, c; \overline{E}). For the “only if” implication, first observe that we have just seen that J\overline{E} = J_{E}. So it suffices to show that s_{T}^{c} (v) ∉ J\overline{E} for all v ∈ Λ^{0} and that if E ⊂ vΛ is finite, does not contain v and does not belong to \overline{E}, then Δ(s_{T}^{c})^{E} ∉ J\overline{E}. The “⊆” containment in Proposition 3.10 and the first assertion of Lemma 3.11 combine to show that s_{T}^{c} (v) ∉ J\overline{E} for each v ∈ Λ^{0}. Likewise, the “⊆” containment in Proposition 3.10 combined with the second assertion of Lemma 3.11 shows that Δ(s_{T}^{c})^{E} ∉ J\overline{E} for all E ∈ FE(Λ) \ \overline{E}. Finally, suppose that F ⊆ vΛ \ {v} is finite and does not belong to FE(Λ). Then there exists λ ∈ vΛ such that Ext(λ; F) = ∅. It is not hard to check (see, for example, the argument of [26, Lemma 4.4]) that \{μ : μμ′ ∈ λS for some μ′\} is an \overline{E}-compatible filter which does not intersect F. So as above, we have ∥π_{T} (Δ(s_{T}^{c})^{F})h_{λ}∥ = 1 for all λ ∈ S and yet another application of the “⊆” containment in Proposition 3.10 implies that Δ(s_{T}^{c})^{F} ∉ J\overline{E}.

Given a finite subset E of Λ and an element μ ∈ E, we write T (E; μ) := {μ′ ∈ s(μ)Λ \ {s(μ)} : μμ′ ∈ E}. Recall from [23] that if E is a finite subset of a finitely aligned k-graph Λ, then there is a finite subset F of Λ which contains E and has the property that if μ, ν, σ, τ ∈ F with d(μ) = d(ν) and d(σ) = d(τ), then μα and τβ belong to F for every να = σβ ∈ MCE(ν, σ). The smallest such set is denoted ΠE and is closed under minimal common extensions (just take μ = ν and τ = σ). Suppose that E = ΠE. The defining property of ΠE ensures that T (E; μ) = T (E; ν) whenever μ, ν ∈ E satisfy d(μ) = d(ν). Fix c ∈ \underline{Z}^{2}(Λ, 𝕋) and a Toeplitz-Cuntz-Krieger (Λ, c)-family t. For μ, ν ∈ E (possibly equal) with d(μ) = d(ν), we write

Θ(t)_{μ,ν}^{E} = t_{μ}Δ(t)^{T (E;μ)}t_{ν}^{∗}.

Lemma 3.12. Let Λ be a finitely aligned k-graph, let c ∈ \underline{Z}^{2}(Λ, 𝕋) and let t be a Toeplitz-Cuntz-Krieger (Λ, c)-family. Suppose that E ⊆ Λ is finite and satisfies E = ΠE. Then M(t)_{E} := span{t_{μ}t_{ν}^{∗} : μ, ν ∈ E, d(μ) = d(ν)} is a finite dimensional C^{∗}-subalgebra of C^{∗}(t), and {Θ(t)_{μ,ν}^{E} : μ, ν ∈ E, d(μ) = d(ν), s(μ) = s(ν), Δ(t)^{T (E;μ)} ≠ 0} is a family of nonzero matrix units spanning M(t)_{E}.

Proof. Using Lemma 3.2, it is easy to see that span{t_{μ}t_{ν}^{∗} : μ, ν ∈ E, d(μ) = d(ν)} is closed under multiplication and hence a subalgebra of C^{∗}(t). Since T (E; μ) = T (E; ν) whenever d(μ) = d(ν) and s(μ) = s(ν), we have (Θ(t)_{μ,ν}^{E})^{∗} = Θ(t)_{ν,μ}^{E}, so M(t)_{E} is a ^{∗}-subalgebra of C^{∗}(t). It is finite-dimensional by definition, and then it is automatically norm-closed and hence a C^{∗}-subalgebra. The argument of [23, Lemma 3.9(ii)], using Lemma 3.7 twice in place of [23, Lemma 3.10] shows that Θ(t)_{μ,ν}^{E}Θ(t)_{η,ξ}^{E} = δ_{ν,η}Θ(t)_{μ,ξ}^{E}. Since Δ(t)^{T (E;μ)} = t_{μ}^{∗}Θ(t)_{μ,ν}^{E}t_{ν} and since Θ(t)_{μ,ν}^{E} = t_{μ}Δ(t)^{T (E;μ)}t_{ν}^{∗}, we have Θ(t)_{μ,ν}^{E} = 0 ⟺ Δ(t)^{T (E;μ)} = 0. The arguments of [23, Proposition 3.5 and Corollary 3.7] only involve products of elements of the form t_{μ}t_{μ}^{∗}, and these satisfy the same relations in a Toeplitz-Cuntz-Krieger (Λ, c)-family as in a Toeplitz-Cuntz-Krieger Λ-family. So the proof of [23, Corollary 3.7] implies that t_{μ}t_{μ}^{∗} = ∑_{μμ′∈E} Δ(t)^{T (E;μμ′)} for μ ∈ E. Now we follow the proof of [23, Corollary 3.11] to see that

t_{μ}t_{ν}^{∗} = ∑_{μα∈E} c(μ, α)\overline{c(ν, α)}Θ(t)_{μα,να}^{E}.  (3.2)

Hence the Θ(t)_{μ,ν}^{E} span M(t)_{E}.

Theorem 3.13. Let Λ be a finitely aligned k-graph, let c ∈ \underline{Z}^{2}(Λ, 𝕋), and let E be a satiated subset of FE(Λ). Suppose that t is a relative Cuntz-Krieger (Λ, c; E)-family. The induced homomorphism π_{t}^{E} : C^{∗}(Λ, c; E) → C^{∗}(t) restricts to an injective homomorphism of C^{∗}(Λ, c; E)^{γ^{c}} if and only if every t_{v} is nonzero, and Δ(t)^{F} ≠ 0 for all F ∈ FE(Λ) \ E.

Proof. Proposition 3.9 implies that if t_{v} = 0 for some v ∈ E^{0} or Δ(t)^{F} = 0 for some F ∈ FE(Λ) \ E, then π_{t}^{E} is not injective on C^{∗}(Λ)^{γ}. Now suppose that each t_{v} ≠ 0 and that Δ(t)^{F} ≠ 0 whenever F ∈ FE(Λ) \ E. Since every finite subset of Λ is contained in a finite subset E satisfying E = ΠE, we have

C^{∗}(Λ, c; E)^{γ^{c}} = \overline{⋃_{ΠE=E} M(s_{E}^{c})^{E}},

and so it suffices to show that π_{t}^{E} is injective (and hence isometric) on each M(s_{E}^{c})^{E}. Fix E such that ΠE = E. Since matrix algebras are simple, Lemma 3.12 implies that it suffices to show that Δ(t)^{T (E;λ)} = 0 implies Δ(s_{E}^{c})^{T (E;λ)} = 0 for each λ ∈ E. We consider two cases. If T (E; λ) ∈ FE(Λ), then Δ(t)^{T (E;λ)} = 0 implies T (E; λ) ∈ E by hypothesis, and then Δ(s_{E}^{c})^{T (E;λ)} = 0 as well. Now suppose that T (E; λ) ∉ FE(Λ). It suffices to show that Δ(t)^{T (E;λ)} ≠ 0. Since T (E; λ) ∉ FE(Λ) and T (E; λ) ∩ Λ^{0} = ∅, the set T (E; λ) is not exhaustive. Fix μ ∈ s(λ)Λ such that MCE(μ, α) = ∅ for every α ∈ T (E; λ). Since t_{μ}^{∗}t_{μ} = t_{s(μ)} is nonzero, q_{μ} = t_{μ}t_{μ}^{∗} is also nonzero. Lemma 3.2 implies that q_{α}q_{μ} = 0 for all α ∈ T (E; λ). Since the q_{η} all commute and q_{μ} is a projection, we deduce that

Δ(t)^{T (E;λ)}q_{μ} = ∏_{α∈T (E;λ)} ((t_{s(λ)} − q_{α})q_{μ}) = q_{μ} ≠ 0.

Hence Δ(t)^{T (E;λ)} ≠ 0.

From this flow a number of uniqueness theorems, all based on the following standard idea from [5].

Lemma 3.14. Let A be a C^{∗}-algebra, Φ : A → A a faithful conditional expectation, and π : A → B a C^{∗}-homomorphism. Suppose that there is a linear map Ψ : B → B such that Ψ ∘ π = π ∘ Φ. Then π is injective if and only if π|_{Φ(A)} is injective.

Proof. The “only if” implication is obvious. For the “if” implication, suppose that π(a) = 0. Then π(Φ(a^{∗}a)) = Ψ(π(a^{∗}a)) = 0. Since π is injective on the range of Φ and Φ is faithful, we deduce that a = 0.

The next result is a version of Coburn’s theorem for T C^{∗}(Λ, c).

Theorem 3.15. Let Λ be a finitely aligned k-graph and let c ∈ \underline{Z}^{2}(Λ, 𝕋). Let t be a Toeplitz-Cuntz-Krieger (Λ, c)-family. Then the induced homomorphism π_{t} of T C^{∗}(Λ, c) is injective if and only if every t_{v} ≠ 0 and Δ(t)^{E} ≠ 0 for every E ∈ FE(Λ).

Proof. Averaging over the gauge action on T C^{∗}(Λ, c) defines a faithful conditional expectation Φ^{γ^{c}} : T C^{∗}(Λ, c) → T C^{∗}(Λ, c)^{γ^{c}} that is characterised by Φ^{γ^{c}}(s_{T}^{c} (μ)s_{T}^{c} (ν)^{∗}) = δ_{d(μ),d(ν)}s_{T}^{c} (μ)s_{T}^{c} (ν)^{∗} for μ, ν ∈ Λ. Lemma 3.12 shows that T C^{∗}(Λ, c)^{γ^{c}} is AF, and \overline{span}\{s_{T}^{c} (μ)s_{T}^{c} (μ)^{∗} : μ ∈ Λ\} is its canonical diagonal subalgebra. So there is a faithful conditional expectation Φ_{D}^{c} : T C^{∗}(Λ, c)^{γ^{c}} → \overline{span}\{s_{T}^{c} (μ)s_{T}^{c} (μ)^{∗} : μ ∈ Λ\} satisfying Φ_{D}^{c}(s_{T}^{c} (μ)s_{T}^{c} (ν)^{∗}) = δ_{μ,ν}s_{T}^{c} (μ)s_{T}^{c} (μ)^{∗}. So Φ := Φ_{D}^{c} ∘ Φ^{γ^{c}} is a faithful conditional expectation satisfying Φ(s_{T}^{c} (μ)s_{T}^{c} (ν)^{∗}) = δ_{μ,ν}s_{T}^{c} (μ)s_{T}^{c} (μ)^{∗} for all μ, ν ∈ Λ. The argument of [22, Proposition 8.9] (this uses Lemmas 8.5–8.8 of the same paper; all the arguments go through for twisted TCK families) gives a normdecreasing linear map Ψ from C^{∗}(t) to \overline{span}\{t_{λ}t_{λ}^{∗} : λ ∈ Λ\} such that Ψ ∘ π_{t} = π_{t} ∘ Φ. Now Lemma 3.14 proves the result.

We also obtain the following version of an Huef and Raeburn’s gauge-invariant uniqueness theorem [11].

Theorem 3.16. Let Λ be a finitely aligned k-graph, let c ∈ \underline{Z}^{2}(Λ, 𝕋), and let E be a satiated subset of FE(Λ). Suppose that t is a relative Cuntz-Krieger (Λ, c; E)-family in a C^{∗}-algebra B and that there is a strongly continuous action β of 𝕋^{k} on B such that β_{z}(t_{λ}) = z^{d(λ)}t_{λ} for all λ ∈ Λ. Then the induced homomorphism π_{t}^{E} : C^{∗}(Λ, c; E) → C^{∗}(t) is injective if and only if every t_{v} is nonzero, and Δ(t)^{F} ≠ 0 for all F ∈ FE(Λ) \ E.

Proof. Lemma 3.14 applied to the expectation Φ obtained from averaging over γ^{c} and the expectation Ψ obtained by averaging over β shows that π_{t}^{E} is injective if and only if it restricts to an injection on C^{∗}(Λ, c; E)^{γ^{c}}. So the result follows from Theorem 3.13.

We also obtain a version of the Cuntz-Krieger uniqueness theorem (cf. [26, Theorem 6.3]). Given a filter S and a path μ with s(μ) ∈ S, we write ℓ_{μ}(S) for the filter {ν ∈ Λ : μS ∩ νΛ ≠ ∅}. We say that a filter S ⊆ Λ is separating if whenever s(μ) = s(ν) ∈ S and there is a filter T such that ℓ_{μ}(S) ∪ ℓ_{ν}(S) ⊆ T, we have μ = ν.

Theorem 3.17. Let Λ be a finitely aligned k-graph, let c ∈ \underline{Z}^{2}(Λ, 𝕋), and let E be a satiated subset of FE(Λ). Suppose that for every v ∈ Λ^{0} there is a separating E-compatible filter S such that r(S) = v, and that for every F ∈ FE(Λ)\E there is a separating E-compatible filter S such that r(S) = r(F) and S ∩ F Λ = ∅. Suppose that t is a relative Cuntz-Krieger (Λ, c; E)-family in a C^{∗}-algebra B. Then the induced homomorphism π_{t}^{E} : C^{∗}(Λ, c; E) → C^{∗}(t) is injective if and only if every t_{v} is nonzero, and Δ(t)^{F} ≠ 0 for all F ∈ FE(Λ)\E.

Proof. Using Remark 3.8, we see that our hypothesis about the existence of separating E-compatible filters is equivalent to Condition (C) of [26]. Now one follows the proof of [26, Theorem 6.3]; the only place where the cocycle c comes up is in the displayed calculation at the top of [26, page 866], where the formula for P_{2}Θ(t)_{λ,μ}^{ΠE}P_{2} picks up a factor of c(λ, x(0, N))\overline{c(μ, x(0, N))}; but the resulting elements still form a family of matrix units, so the rest of the argument proceeds without change.

The hypothesis of Theorem 3.17 simplifies significantly in the key case where E = FE(Λ). Recall from [18, Definition 3.3] that a finitely aligned k-graph Λ is cofinal if, for all v, w ∈ Λ^{0} there exists a finite exhaustive subset E of vΛ (here E may contain v) such that wΛs(α) ≠ ∅ for every α ∈ E. Recall from [18] that a k-graph Λ is aperiodic if whenever μ, ν are distinct paths with the same source, there exists τ ∈ s(μ)Λ such that MCE(μτ, ντ) = ∅.

Corollary 3.18. Let Λ be a finitely aligned k-graph and let c ∈ \underline{Z}^{2}(Λ, 𝕋). Suppose Λ is aperiodic. Then a homomorphism π : C^{∗}(Λ, c) → B is injective if and only if every π(s_{v}^{c}) ≠ 0. If Λ is cofinal then C^{∗}(Λ, c) is simple.

Proof. Condition (A) of [10], reinterpreted using Remark 3.8, requires that for every v ∈ Λ^{0} there is a separating FE(Λ)-compatible filter S containing v. So the implication (i) ⟹ (iii) of [18, Proposition 3.6] implies that the hypotheses of Theorem 3.17 are satisfied with E = FE(Λ). This proves the first assertion. Now suppose that Λ is cofinal. Then the argument of (ii) ⟹ (iii) of [18, Theorem 5.1] carries over unchanged to the twisted setting to show that if I is an ideal of C^{∗}(Λ, E) and s_{v}^{c} ∈ I for some v, then s_{w}^{c} ∈ I for every w ∈ Λ^{0} and hence I = C^{∗}(Λ, c). So C^{∗}(Λ, c) is simple by the preceding paragraph.

Recall from [8] that if Λ is a finitely aligned k-graph, then a generalised cycle in Λ is a pair (μ, ν) ∈ Λ such that r(μ) = r(ν), s(μ) = s(ν) and MCE(μτ, ν) ≠ ∅ for all τ ∈ s(μ)Λ. We say that a generalised cycle (μ, ν) has an entrance if there exists τ ∈ s(ν)Λ such that MCE(μ, ντ) = ∅. If c ∈ \underline{Z}^{2}(Λ, 𝕋) and t is a Toeplitz-Cuntz-Krieger (Λ, c)-family such that t_{s(τ)} ≠ 0, then V := t_{μ}t_{ν}^{∗} satisfies q_{μ} = V V^{∗} ∼ V^{∗}V = q_{ν} > q_{ν} − q_{ντ} ≥ q_{μ}, where ∼ denotes Murray-von Neumann equivalence. Hence q_{μ} is infinite.

Proposition 3.19. Let Λ be a finitely aligned k-graph with no sources, and let c ∈ \underline{Z}^{2}(Λ, 𝕋). Suppose that Λ is aperiodic and that for every v ∈ Λ^{0} there is a generalised cycle (μ, ν) with an entrance such that vΛr(μ) ≠ ∅. Then every hereditary subalgebra of C^{∗}(Λ, c) contains an infinite projection. If Λ is cofinal, then C^{∗}(Λ, c) is simple and purely infinite.

Proof. Let v ∈ Λ^{0} and choose a generalised cycle (μ, ν) with an entrance such that vΛr(μ) ≠ ∅; say λ ∈ vΛr(μ). Then s^{c}(v) ≥ s^{c}(λ)s^{c}(λ)^{∗} ∼ s^{c}(λ)^{∗}s^{c}(λ) ≥ p^{c}(μ) is infinite. Using this in place of [25, Lemma 8.13], we can now follow the proof of [25, Proposition 8.8] (including the proof of [25, Lemma 8.12]).

4 Gauge-invariant ideals

In this section we list the gauge-invariant ideals of each C^{∗}(Λ, c; E). There is by now a fairly standard program for this (the basic idea goes back to [6] and [11]). The standard arguments are applied to the untwisted Cuntz-Krieger algebras of finitely aligned k-graphs in [25], and we follow the broad strokes of that treatment. The twist does not affect the arguments much, but we need to make a few adjustments to pass from C^{∗}(Λ, c) to C^{∗}(Λ, c; E). So we give more detail here than in the preceding section.

Definition 4.1. Let Λ be a finitely aligned k-graph and let E be a satiated subset of FE(Λ). We say that H ⊆ Λ^{0} is hereditary if s(HΛ) ⊆ H, and that it is E-saturated if whenever E ∈ E and s(E) ⊆ H we have r(E) ∈ H.

Recall from [25, Lemma 4.1] that if Λ is a finitely aligned k-graph and H ⊆ Λ^{0} is hereditary, then Λ \ ΛH is a finitely aligned k-graph under the operations and degree map inherited from Λ.

Lemma 4.2. Let Λ be a finitely aligned k-graph and let E be a satiated subset of FE(Λ). Suppose that H ⊆ Λ^{0} is hereditary and E-saturated. Then

E_{H} := {E \ EH : E ∈ E, r(E) ∉ H}

is a subset of FE(Λ \ ΛH).

Proof. No element F of E_{H} contains r(F) because no element E of E contains r(E). The elements of E_{H} are nonempty because H is E-saturated. Fix F ∈ E_{H}, say r(F) = v, and fix λ ∈ vΛ \ ΛH. Choose E ∈ E such that F = E \ EH. We have to show that there exists μ ∈ F such that MCE(μ, λ) ∩ (Λ \ ΛH) ≠ ∅. We consider two cases. First suppose that λ ∈ EΛ, say λ = μμ′ with μ ∈ E. Since H is hereditary and s(λ) = s(μ′) ∉ H, we have s(μ) = r(μ′) ∉ H, so μ ∈ F satisfies λ ∈ MCE(μ, λ) ∩ (Λ \ ΛH). Now suppose that λ ∉ EΛ. Then Ext(λ; E) ∈ E by (S2). Since H is E-saturated and s(λ) ∉ H, it follows that s(Ext(λ; E)) ⊈ H. So there exists μ ∈ E and μα = λβ ∈ MCE(μ, λ) with s(α) ∉ H. Since H is hereditary, it follows that s(μ) ∉ H, so μ ∈ F and MCE(μ, λ) ∩ (Λ \ ΛH) ≠ ∅.

Lemma 4.3. Let Λ be a finitely aligned k-graph, let c ∈ \underline{Z}^{2}(Λ, 𝕋), and let E be a satiated subset of FE(Λ). Let I be an ideal of C^{∗}(Λ, c; E). Then

1. H_{I} := {v ∈ Λ^{0} : s_{E}^{c}(v) ∈ I} is hereditary and E-saturated;

2. B_{I} := {F ∈ FE(Λ\ΛH_{I}) : Δ(s_{E}^{c})^{F} ∈ I} is a satiated subset of FE(Λ\ΛH); and

3. E_{H_{I}} ⊆ B_{I}.

Proof. (1). If r(λ) ∈ H_{I}, then s_{E}^{c}(s(λ)) = s_{E}^{c}(λ)^{∗}s_{E}^{c}(r(λ))s_{E}^{c}(λ) ∈ I, and hence H_{I} is hereditary. To see that it is E-saturated, suppose that E ∈ E and s(E) ⊆ H, and let v = r(E). The calculation of [22, Proposition 8.6] implies that s_{E}^{c}(v) = Δ(s_{E}^{c})^{∨E} + ∑_{μ∈∨E} s_{E}^{c}(μ)Δ(s_{E}^{c})^{T (∨E;μ)}s_{E}^{c}(μ)^{∗}. Since E ∈ E, condition (S1) implies that ∨E ∈ E, and so Δ(s_{E}^{c}) = 0. Since H is hereditary, each s_{E}^{c}(μ)Δ(s_{E}^{c})^{T (∨E;μ)}s_{E}^{c}(μ)^{∗} ≤ s_{E}^{c}(μ)s_{E}^{c}(s(μ))s_{E}^{c}(μ)^{∗} ∈ I, giving s_{E}^{c}(v) ∈ I. (2). Write Γ := Λ \ ΛH, and fix F ∈ B_{I}, so that Δ(s_{E}^{c})^{F} ∈ I. To see that B_{I} satisfies (S1), fix λ ∈ r(F)Γ \ {r(F)}. Then Δ(s_{E}^{c})^{F ∪{λ}} = Δ(s_{E}^{c})^{F} (p_{E}^{c}(r(F)) − p_{E}^{c}(λ)) ∈ I. For (S2), fix F ∈ B_{I} and λ ∈ r(F)Γ \ F Γ. Then Lemma 3.7 gives

Δ(s_{E}^{c})^{Ext(λ;F)} = s_{E}^{c}(λ)^{∗}Δ(s_{E}^{c})^{F} s_{E}^{c}(λ) ∈ I.

For (S3), suppose λ, λλ′ ∈ F with λ′ ≠ s(λ). Then p_{E}^{c}(r(λ)) − p_{E}^{c}(λλ′) ≥ p_{E}^{c}(r(F)) − p_{E}^{c}(λ) ≥ Δ(s_{E}^{c})^{F \λλ′}, so Δ(s_{E}^{c})^{F \λλ′} = Δ(s_{E}^{c})^{F \λλ′}(p_{E}^{c}(r(F)) − p_{E}^{c}(λλ′)) = Δ(s_{E}^{c})^{F} ∈ I. For (S4), suppose that λ ∈ F and G ∈ s(λ)B_{I}. Let F′ := F \ {λ} ∪ λG. For α ∈ G, both p_{E}^{c}(r(F)) and p_{E}^{c}(λ) dominate (p_{E}^{c}(λ) − p_{E}^{c}(λα)), giving (p_{E}^{c}(r(F)) − p_{E}^{c}(λ))(p_{E}^{c}(λ) − p_{E}^{c}(λα)) = 0. Hence

∏_{α∈G}(p_{E}^{c}(r(F)) − p_{E}^{c}(λα))

= ∏_{α∈G} ((p_{E}^{c}(r(F)) − p_{E}^{c}(λ)) + (p_{E}^{c}(λ) − p_{E}^{c}(λα)))

= ∑_{H⊆G} (p_{E}^{c}(r(F)) − p_{E}^{c}(λ))^{|H|} ∏_{α∈G\H} (p_{E}^{c}(λ) − p_{E}^{c}(λα))

= (p_{E}^{c}(r(F)) − p_{E}^{c}(λ)) + ∏_{α∈G}(p_{E}^{c}(λ) − p_{E}^{c}(λα)).

Hence

Δ(s_{E}^{c})^{F′} = Δ(s_{E}^{c})^{F \{λ}}(p_{E}^{c}(r(F)) − p_{E}^{c}(λ)) + Δ(s_{E}^{c})^{F \{λ}} ∏_{α∈G}(p_{E}^{c}(λ) − p_{E}^{c}(λα)).

We have Δ(s_{E}^{c})^{F \{λ}}(p_{E}^{c}(r(F))−p_{E}^{c}(λ)) = Δ(s_{E}^{c})^{F} ∈ I, and a quick computation using that range projections commute shows that ∏_{α∈G}(p_{E}^{c}(λ) − p_{E}^{c}(λα)) = s_{E}^{c}(λ)Δ(s_{E}^{c})^{G}s_{E}^{c}(λ)^{∗} ∈ I. Hence Δ(s_{E}^{c})^{F′} ∈ I.

(3). Fix F ∈ E_{H_{I}}, and choose E ∈ E with r(E) ∉ H such that F = E \ EH. Then Δ(s_{E}^{c})^{E} = 0. Let q_{I} be the quotient map a ↦ a + I. For μ ∈ EH, we have q_{I}(p_{E}^{c}(μ)) = 0, and so q_{I}(p_{E}^{c}(r(E)) − p_{E}^{c}(μ)) = q_{I}(p_{E}^{c}(r(E)). Hence

0 = q_{I}(Δ(s_{E}^{c})^{E}) = ∏_{μ∈E} (q_{I}(p_{E}^{c}(r(E)) − p_{E}^{c}(μ)))

= ∏_{μ∈F} (q_{I}(p_{E}^{c}(r(E)) − p_{E}^{c}(μ))) ∏_{μ∈EH} q_{I}(p_{E}^{c}(r(E)) = q_{I}(Δ(s_{E}^{c})^{F}).

So Δ(s_{E}^{c})^{F} ∈ I and hence F ∈ B_{I}.

Suppose that Λ is a finitely aligned k-graph, that c ∈ \underline{Z}^{2}(Λ, 𝕋) and that E ⊆ FE(Λ) is satiated. Suppose that H ⊆ Λ^{0} is hereditary and E-saturated and that B ⊆ FE(Λ \ ΛH) is satiated and contains E_{H}. We write I_{H,B}^{c} for the ideal of C^{∗}(Λ, c; E) generated by {s_{E}^{c}(v) : v ∈ H} ∪ {Δ(s_{E}^{c})^{E} : E ∈ B}. Observe that the cocycle c restricts to a cocycle, which we also denote c, on the subgraph Λ \ ΛH.

Theorem 4.4. Suppose that Λ is a finitely aligned k-graph, that c ∈ \underline{Z}^{2}(Λ, 𝕋) and that E ⊆ FE(Λ) is satiated. Suppose that H ⊆ Λ^{0} is hereditary and Esaturated and that B ⊆ FE(Λ \ ΛH) is satiated and contains E_{H}. There is a homomorphism π_{H,B} : C^{∗}(Λ, c; E) → C^{∗}(Λ \ ΛH, c; B) such that

π_{H,B}(s_{E}^{c}(λ)) = \begin{cases}s_{B}^{c}(λ) & if s(λ) ∉ H \\ 0 & if s(λ) ∈ H. \end{cases}  (4.1)

We have ker(π_{H,B}) = I_{H,B}, and H = H_{I_{H,B}} and B = B_{I_{H,B}}.

Proof. Routine calculations show that the formula for π_{H,B} defines a Toeplitz-Cuntz-Krieger (Λ, c)-family in C^{∗}(Λ \ ΛH, c; B). That E_{H} ⊆ B ensures that this family also satisfies (CK). Hence the universal property of C^{∗}(Λ, c; E) gives a homomorphism π_{H,B} satisfying (4.1), which is surjective because its image contains the generators of C^{∗}(Λ \ ΛH, c; B). To see that ker(π_{H,B}) = I_{H,B}, observe that the generators of I_{H,B} belong to ker(π_{H,B}), and so π_{H,B} descends to a homomorphism π̃ : C^{∗}(Λ, c; E)/I_{H,B} → C^{∗}(Λ \ ΛH, c; B). It suffices to show that π̃ is injective, and we do this by constructing an inverse for π̃. Define elements {t_{λ} : λ ∈ Λ \ ΛH} in C^{∗}(Λ, c; E)/I_{H,B} by t_{λ} = s_{E}^{c}(λ) + I_{H,B}. Since the relations (TCK1)–(TCK3) for Λ \ ΛH families hold in any Toeplitz-Cuntz-Krieger Λ family, t satisfies (TCK1)–(TCK3). For λ, μ ∈ Λ \ ΛH, it is straightforward to check that MCE_{Λ\ΛH}(λ, μ) = MCE_{Λ}(λ, μ) \ ΛH. If η ∈ MCE_{Λ}(λ, μ) ∩ ΛH, then q_{η} is the zero element of C^{∗}(Λ, c; E)/I_{H,B}, and it follows that the t_{λ} satisfy (TCK4). They satisfy (CK) because E ∈ B implies Δ(s_{E}^{c})^{E} ∈ I_{H,B} so that Δ(t)^{E} = 0. Now the universal property of C^{∗}(Λ \ ΛH, c; B) provides a homomorphism π_{t} : C^{∗}(Λ \ ΛH, c; B) → C^{∗}(Λ, c; E)/I_{H,B}. The map π_{t} is an inverse for π̃ on generators, and so π̃ is injective.

For the last assertion, observe that we have H ⊆ H_{I_{H,B}} and B ⊆ B_{I_{H,B}} by definition. For the reverse inclusions, observe that if v ∉ H then Proposition 3.9 applied to C^{∗}(Λ \ ΛH, c; B) implies that π_{H,B}(s_{E}^{c}(v)) ≠ 0, so that v ∉ H_{I_{H,B}}. Similarly if E ∈ FE(Λ \ ΛH) \ B, then Proposition 3.9 implies that π_{H,B}(Δ(s_{E}^{c})^{E}) ≠ 0 and hence E ∉ B_{I_{H,B}}.

Corollary 4.5. With the hypotheses of Theorem 4.4, the homomorphism π_{H,B} descends to an isomorphism ϕ_{H,B} : C^{∗}(Λ, c; E)/I_{H,B} → C^{∗}(Λ \ ΛH, c; B) such that

ϕ_{H,B}(s_{E}^{c}(λ) + I_{H,B}) = s_{B}^{c}(λ) for all λ ∈ Λ \ ΛH.

We are now ready to give a listing of the gauge-invariant ideals of C^{∗}(Λ, c; E). Let Λ be a finitely aligned k-graph and E a satiated subset of FE(Λ). We define SH_{E}×S to be the collection of all pairs (H, B) such that H ⊆ Λ^{0} is hereditary and E-saturated, and B ⊆ FE(Λ \ ΛH) is satiated and satisfies E_{H} ⊆ B.

Theorem 4.6. Let Λ be a finitely aligned k-graph, E a satiated subset of FE(Λ), and c ∈ \underline{Z}^{2}(Λ, 𝕋). The map (H, B) ↦ I_{H,B} is a bijection from SH_{E}×S to the collection of gauge-invariant ideals of C^{∗}(Λ, c; E). We have I_{H_{1},B_{1}} ⊆ I_{H_{2},B_{2}} if and only if both of the following hold: H_{1} ⊆ H_{2}; and whenever E ∈ B_{1} and r(E) ∉ H_{2}, we have E \ EH_{2} ∈ B_{2}.

Proof. The final statement of Theorem 4.4 implies that (H, B) ↦ I_{H,B} is injective. To see that it is surjective, fix a gauge-invariant ideal I of C^{∗}(Λ, c; E). We must show that I = I_{H_{I},B_{I}}. We clearly have I_{H_{I},B_{I}} ⊆ I, and so the quotient map q_{I} : C^{∗}(Λ, c; E) → C^{∗}(Λ, c; E)/I determines a homomorphism q̃_{I} : C^{∗}(Λ, c; E)/I_{H_{I},B_{I}} → C^{∗}(Λ, c; E)/I. We must show that q̃_{I} is injective. Let ϕ_{H_{I},B_{I}} : C^{∗}(Λ, c; E)/I_{H_{I},B_{I}} → C^{∗}(Λ \ ΛH_{I}, c; B_{I}) be the isomorphism of Corollary 4.5, and let θ := q̃_{I} ∘ ϕ_{H_{I},B_{I}}. Since I is gauge-invariant, the gauge action on C^{∗}(Λ, c; E) descends to an action β on C^{∗}(Λ, c; E)/I such that β_{z} ∘θ = θ ∘γ_{z} for all z. For each v ∈ (Λ \ ΛH_{I})^{0} we have s_{E}^{c}(v) ∉ I by definition of H_{I}, and so each θ(s_{B_{I}}^{c} (v)) ≠ 0. Similarly, if F ∈ FE(Λ \ ΛH) \ B_{I}, then Δ(s_{E}^{c})^{F} ∉ I by definition of B_{I}, and so θ(Δ(s_{E}^{c})^{F}) ≠ 0. So the gauge-invariant uniqueness theorem (Theorem 3.16) implies that θ is injective. Hence q̃_{I} is injective. The proof of the final statement is identical to that of [25, Theorem 6.2].

We are now ready to prove the ⊇ containment from Proposition 3.10 as promised in Section 3.

Proof of ⊇ in Proposition 3.10. Let

J := \{a ∈ T C^{∗}(Λ, c) : lim_{λ∈S} π_{T} (a)h_{λ} = 0 for every \overline{E}-compatible filter S\}.

This J is clearly a linear subspace of T C^{∗}(Λ), and an ε/3-argument shows that it is norm-closed. If S is an \overline{E}-compatible filter and s(λ) = r(S), then λS is a cofinal subset of the \overline{E}-compatible filter ℓ_{λ}(S). So if a ∈ J, then lim_{μ∈S} π_{T} (as_{λ})h_{μ} = lim_{ν∈λS} π_{T} (a)h_{ν} = 0, giving aT_{λ} ∈ J. Similarly, since ℓ_{λ}^{∗}(S) is an E-compatible filter whenever S is, for a ∈ J and λ ∈ Λ we have lim_{μ∈S} π_{T} (as_{λ}^{∗})h_{μ} = lim_{ν∈ℓ_{λ}^{∗}(S)} π_{T} (a)h_{ν} = 0, so that as_{λ}^{∗} ∈ J. Clearly T_{λ}a and T_{λ}^{∗}a belong to J. So J is an ideal of T C^{∗}(Λ). Proposition 3.10 implies that Δ(s_{T}^{c})^{E} ∈ J for all E ∈ E. Lemma 3.11 implies that s_{T}^{c} (v) ∉ π_{T} (J) for all v ∈ Λ^{0} and that Δ(s_{T}^{c})^{E} ∉ J for E ∈ FE(Λ) \ E. So H_{J} = ∅ = H_{J_{E}}, and B_{J} = E = B_{J_{E}}. So the result will follow from Theorem 4.6 once we establish that J is gauge-invariant. Fix a ∈ J and z ∈ 𝕋^{k}; we must show that γ_{z}(a) ∈ J. There is a unitary U_{z} in B(ℓ^{2}(Λ)) given by U_{z}(h_{λ}) = z^{d(λ)}h_{λ}. We have U_{z}T_{λ}U_{z}^{∗} = z^{d(λ)}T_{λ} for all z, λ, and so Ad U_{z} ∘ π_{T} = π_{T} ∘ γ_{z}. Fix an \overline{E}-compatible filter S. For λ ∈ S, we have

∥π(γ_{z}(a))h_{λ}∥ = ∥U_{z}π(a)U_{z}^{∗}h_{λ}∥ ≤ ∥U_{z}∥∥π(a)\overline{z}^{d(λ)}h_{λ}∥ = ∥π(a)h_{λ}∥.

So lim_{λ∈S} ∥π(γ_{z}(a))h_{λ}∥ ≤ lim_{λ∈S} ∥π(a)h_{λ}∥ = 0, giving γ_{z}(a) ∈ J.

5 Nuclearity and the Universal Coefficient Theorem

We show that each C^{∗}(Λ, c; E) is nuclear and satisfies the Universal Coefficient Theorem. Given a set X we write K_{X} for the C^{∗}-algebra generated by nonzero matrix units {θ_{x,y} : x, y ∈ X}.

Lemma 5.1. Let Λ be a finitely aligned k-graph and let E be a satiated subset of FE(Λ). Suppose that b : Λ^{0} → ℤ^{k} satisfies d(λ) = b(s(λ)) − b(r(λ)) for all λ ∈ Λ. Let c ∈ \underline{Z}^{2}(Λ, 𝕋) and let t be a relative Cuntz-Krieger (Λ, c; E)-family with each t_{v} ≠ 0.

1. For n ∈ ℕ^{k} there is an isomorphism \overline{span}\{t_{μ}t_{ν}^{∗} : b(s(μ)) = b(s(ν)) = n} ≅ ⨁_{b(v)=n} K_{Λv} which carries each t_{μ}t_{ν}^{∗} to θ_{μ,ν}.

2. If s(μ) = s(ν) and s(η) = s(ζ), then t_{μ}t_{ν}^{∗}t_{η}t_{ζ}^{∗} ∈ \overline{span}\{t_{α}t_{β}^{∗} : b(s(α)) = b(s(β)) = b(s(μ)) ∨ b(s(ν))}.

3. If N ⊆ ℕ^{k} is finite and m, n ∈ N implies m ∨ n ∈ N, then B(t)_{N} := \overline{span}\{t_{μ}t_{ν}^{∗} : b(s(μ)) = b(s(ν)) ∈ N\} is an AF algebra.

Proof. (1). We just have to check that the t_{μ}t_{ν}^{∗} are matrix units. So suppose that s(μ) = s(ν) = v and s(η) = s(ζ) = w and b(v) = b(w) = n. Then

t_{μ}t_{ν}^{∗}t_{η}t_{ζ}^{∗} = ∑_{να=ηζ∈MCE(ν,η)} c(μ, α)\overline{c(ν, α)}c(ζ, β)\overline{c(η, β)}t_{μα}t_{ζβ}^{∗}.

If r(ν) ≠ r(η), then MCE(ν, η) = ∅ and so t_{μ}t_{ν}^{∗}t_{η}t_{ζ}^{∗} = 0. If r(ν) = r(η), then we have d(ν) = b(s(ν)) − b(r(ν)) = n − b(r(ν)) and similarly d(η) = n − b(r(η)) so that d(ν) = d(η), giving t_{μ}t_{ν}^{∗}t_{η}t_{ζ}^{∗} = δ_{ν,η}t_{μ}t_{ν}^{∗}t_{ν}t_{ζ}^{∗} = δ_{ν,η}t_{μ}t_{ζ}^{∗}. (2). Suppose that b(s(ν)) = m and b(s(η)) = n, and that λ ∈ MCE(ν, η). Since MCE(ν, η) ≠ ∅, we have r(ν) = r(η), and in particular b(r(ν)) = b(r(η)) = p, say. Thus m − d(ν) = p = n − d(η). We have d(λ) = d(μ) ∨ d(ν) = (m ∨ n) − p and so b(s(λ)) = b(r(λ)) + (d(μ) ∨ d(ν)) = m ∨ n. So the result follows from Lemma 3.2. (3). The proof is by induction on |N|. For |N| = 1, the result follows from part (1). Fix N with |N| ≥ 2, and suppose as an inductive hypothesis that B(t)_{M} is AF whenever |M| < |N|. Pick a minimal n ∈ N and observe that M := N \ {n} is closed under ∨, so that B_{M} is AF. Part 2 implies that B_{M} is an ideal of B_{N} and B_{N}/B_{M} is a quotient of B_{{n}} and therefore is AF. Since extensions of AF algebras by AF algebras are AF, the result follows.

Recall that if Λ is a finitely aligned k-graph, then Λ ×_{d} ℤ^{k} is the skew-product k-graph which is equal as a set to Λ × ℤ^{k} and has structure maps r(λ, n) = (r(λ), n), s(λ, n) = (s(λ), n + d(λ)), (λ, n)(μ, n + d(λ)) = (λμ, n), and d(λ, n) = d(λ). For c ∈ \underline{Z}^{2}(Λ, 𝕋), the map c × 1 : ((λ, n), (μ, n + d(λ))) ↦ c(λ, μ) belongs to \underline{Z}^{2}(Λ ×_{d} ℤ^{k}, 𝕋).

Corollary 5.2. Let Λ be a finitely aligned k-graph, E a satiated subset of FE(Λ), and c ∈ \underline{Z}^{2}(Λ, 𝕋). Then C^{∗}(Λ, c; E) is Morita equivalent to the crossedproduct of an AF algebra by ℤ^{k}. In particular, it belongs to the bootstrap class N of [24], and so is nuclear and satisfies the UCT.

Proof. Let Γ := Λ ×_{d} ℤ^{k}, let c′ := c × 1 and let F := E × ℤ^{k}. The argument of [25, Lemma 8.3] shows that F ⊆ FE(Γ) and that the crossed-product of C^{∗}(Λ, c; E) by the gauge action is isomorphic to C^{∗}(Γ, c′; F). Since the subalgebras B_{N} of C^{∗}(Γ, c′; F) described in Lemma 5.1 are AF algebras and satisfy C^{∗}(Γ, c′; F) = \overline{⋃_{N} B_{N}}, and since the class of AF algebras is closed under direct limits, C^{∗}(Γ, c′; F) is AF. Now Takai duality implies that C^{∗}(Λ, c; E) is Morita equivalent to a crossed product of an AF algebra by ℤ^{k}, and the result follows.

6 K-theory

In this section we follow the program of [17] to show that if c has the form c(μ, ν) = e^{iω(μ,ν)} for some ω ∈ \underline{Z}^{2}(Λ, ℝ), then the K-theory of C^{∗}(Λ, c; E) is isomorphic to that of C^{∗}(Λ; E).

Theorem 6.1. Let Λ be a finitely aligned k-graph and suppose that E ⊆ FE(Λ) is satiated. Suppose that ω ∈ \underline{Z}^{2}(Λ, ℝ), and define c ∈ \underline{Z}^{2}(Λ, 𝕋) by c(μ, ν) = e^{iω(μ,ν)}. Then C^{∗}(Λ, c; E) is unital if and only if Λ^{0} is finite. There is an isomorphism: K_{∗}(C^{∗}(Λ, c; E) ≅ K_{∗}(C^{∗}(Λ; E)) taking [s_{E}^{c}(v)] to [s_{E}(v)] for each v ∈ Λ^{0}, and taking [1_{C^{∗}(Λ,c;E)}] to [1_{C^{∗}(Λ;E)}] if Λ^{0} is finite.

The proof of Theorem 6.1 appears at the end of the section; we have to do some preliminary work first. Following [17, Definition 2.1], given a finitely aligned k-graph Λ, a locally compact abelian group A and a cocycle ω ∈ \underline{Z}^{2}(Λ, A), a Toeplitz c-representation of (Λ, A) on a C^{∗}-algebra B consists of a map ϕ : Λ → M(B) and a homomorphism π : C^{∗}(A) → M(B) such that ϕ(λ)π(f) ∈ B for all λ ∈ Λ and f ∈ C^{∗}(A), and such that (R1) π(f)ϕ(λ) = ϕ(λ)π(f) for all λ ∈ Λ and f ∈ C^{∗}(A);

(R2) {ϕ(v) : v ∈ Λ^{0}} is a set of mutually orthogonal projections and ∑_{v∈Λ^{0}} ϕ(v) → 1 strictly in M(B);

(R3) ϕ(λ)ϕ(μ) = π(ω(λ, μ))ϕ(λμ) whenever s(λ) = r(μ); and

(R4) the ϕ(λ) satisfy (TCK3) and (TCK4).

If the ϕ(λ) satisfy relation (CK) with respect to a satiated subset E of FE(Λ), then (ϕ, π) is an E-relative c-representation of (Λ, A). Calculations identical to those of Lemma 3.2 show that given a Toeplitz crepresentation of (Λ, A), the C^{∗}-algebra C^{∗}(ϕ, π) := C^{∗}{ϕ(λ)π(f) : f ∈ C^{∗}(A), λ ∈ Λ} is spanned by {ϕ(μ)π(f)ϕ(ν)^{∗} : s(μ) = s(ν), f ∈ C^{∗}(A)}, and that

ϕ(ν)^{∗}ϕ(η) = ∑_{να=ηβ∈MCE(ν,η)} ϕ(α)π(ω(η, β))π(ω(ν, α))^{∗}ϕ(β)^{∗},

where we are identifying elements of A with the corresponding multiplier unitaries of C^{∗}(A). There is a universal C^{∗}-algebra C^{∗}(Λ, A, ω; E) for Erelative c-representations of (Λ, A), and we denote the universal E-relative representation by (ι_{Λ}^{E,ω}, ι_{A}^{E,ω}). The argument of [17, Lemma 2.4] shows that C^{∗}(Λ, A, ω; E) is canonically isomorphic to C^{∗}(Λ, A, ω′; E) if ω and ω′ are cohomologous. Let Â denote the Pontryagin dual of A. As in Proposition 2.5 of [17], the algebra C^{∗}(Λ, A, ω; E) is a C_{0}(Â)-algebra with respect to the inclusion i_{A}^{E,ω} : C^{∗}(A) ≅ C_{0}(Â) ↪ ZM(C^{∗}(Λ, A, ω; E)), and for each χ ∈ Â there is a homomorphism π_{χ} : C^{∗}(Λ, A, ω; E) → C^{∗}(Λ, χ ∘ ω; E) satisfying π_{χ}(ι_{Λ}^{E,ω}(λ)ι_{A}^{E,ω}(f)) = f(χ)s_{E}^{c}(λ). Moreover, π_{χ} descends to an isomorphism of the fibre C^{∗}(Λ, A, ω; E)_{χ} with C^{∗}(Λ, χ ∘ ω; E). When ω ≡ 1, the universal properties of C^{∗}(Λ, A, 1, E) and C^{∗}(Λ, 1; E)⊗C^{∗}(A) give an isomorphism C^{∗}(Λ, A, 1; E) ≅ C^{∗}(Λ, 1; E) ⊗ C^{∗}(A). We now consider a finitely aligned k-graph Γ endowed with a map b : Γ^{0} → ℤ^{k} such that d(λ) = b(s(λ)) − b(r(λ)) for all λ ∈ Γ. Our application is when Γ is the skew-product Λ ×_{d} ℤ^{k} as in the preceding section, but it will keep our notation simpler to deal with the general situation. The proof of Lemma 8.2 of [25] shows that given a finite subset E of Γ, there is a minimal finite Ẽ ⊆ Γ such that E ⊆ Ẽ and whenever μ, ν, η, ζ ∈ Ẽ with s(μ) = s(ν), s(η) = s(ζ) and να = ηβ ∈ MCE(ν, η), we have μα, ζβ ∈ Ẽ. (This set plays a similar role to the set ΠE used to examine the core in Section 3, and is constructed in a similar way.) The set Ẽ satisfies Ẽ = ∨Ẽ and has the property that T (Ẽ; μ) = T (Ẽ; ν) whenever μ, ν ∈ Ẽ and s(μ) = s(ν). Let ω ∈ \underline{Z}^{2}(Γ, A) and let (ϕ, π) be a Toeplitz ω-representation of (Γ, A). For v ∈ Λ^{0} and a finite E ⊆ vΓ, let Δ(ϕ)^{E} := ∏_{λ∈E} ϕ(v) − ϕ(λ)ϕ(λ)^{∗}. Since the π(ω(μ, ν)) are unitaries and each π(ω(r(μ), μ)) = 1_{M(B)}, the calculations of Lemma 3.7 show that

Δ(ϕ)^{E}ϕ(μ) = ϕ(μ)Δ(ϕ)^{Ext(μ;E)}.  (6.1)

The ϕ(μ)ϕ(μ)^{∗} satisfy the same commutation relations as the t_{μ}t_{μ}^{∗} in a Toeplitz-Cuntz-Krieger family. So if E = Ẽ, then since Ẽ = ∨Ẽ the argument of [23, Corollary 3.7] gives

ϕ(λ)ϕ(λ)^{∗} = ∑_{λλ′∈E} Δ(ϕ)^{T (E;λλ′)}.  (6.2)

The argument of Lemma 3.12 shows that Θ(ϕ)_{μ,ν}^{E} := ϕ(μ)Δ(ϕ)^{T (E;μ)}ϕ(ν)^{∗} defines matrix units, and that for μ, ν ∈ E with s(μ) = s(ν), we have

ϕ(μ)ϕ(ν)^{∗} = ∑_{μα∈E} π(ω(μ, α))π(ω(ν, α))^{∗}Θ(ϕ)_{μα,να}^{E}.  (6.3)

Lemma 6.2. Let Γ be a finitely aligned k-graph and suppose that b : Λ^{0} → ℤ^{k} satisfies d(λ) = b(s(λ))−b(r(λ)) for all λ. Suppose that F ⊆ FE(Γ) is satiated. Let A be a locally compact abelian group and consider ω ∈ \underline{Z}^{2}(Γ, A). If E = Ẽ ⊆ Γ then

M_{E}^{F,ω} := span{ι_{Γ}^{F,ω}(μ)ι_{A}^{F,ω}(f)ι_{Γ}^{F,ω}(ν)^{∗} : μ, ν ∈ E, f ∈ C^{∗}(A)}
= span{Θ(ι_{Γ}^{F,ω})_{μ,ν}^{E}ι_{A}^{F,ω}(f) : μ, ν ∈ E, f ∈ C^{∗}(A)},

and there is an isomorphism of M_{E}^{F,ω} onto ⨁_{v∈s(E)} M_{Ev}(ℂ) ⊗ C^{∗}(A) that carries each spanning element Θ(ι_{Γ}^{F,ω})_{μ,ν}^{E}ι_{A}^{F,ω}(f) to θ_{μ,ν} ⊗ f.

Proof. Equation (6.3) establishes the displayed equation. We saw above that the Θ(ι_{Γ}^{F,ω})_{μ,ν}^{E} are matrix units, and they commute with the ι_{A}^{F,ω}(f) because the range of ι_{A}^{F,ω} is central. Now we follow the argument of Lemma 4.1 of [17]: The universal property of ⨁_{v∈s(E)} M_{Ev}(ℂ)⊗C^{∗}(A) gives a surjection ψ : ⨁_{v∈s(E)} M_{Ev}(ℂ)⊗C^{∗}(A) → M_{E}^{F,ω} such that ψ(θ_{μ,ν} ⊗f) = Θ(ι_{Γ}^{F,ω})_{μ,ν}^{E}ι_{A}^{F,ω}(f). For each χ ∈ Â, and each f ∈ Â such that f(χ) = 1, the canonical homomorphism π_{χ} : C^{∗}(Γ, A, ω; F) → C^{∗}(Γ, χ ∘ ω; E) carries the Θ(ι_{Γ}^{E,ω})_{μ,ν}^{E}ι_{A}^{E,ω}(f) to the matrix units θ(s_{E}^{χ∘ω})_{μ,ν}^{E}, and Lemma 3.12 shows that any given θ(s_{E}^{χ∘ω})_{μ,ν}^{E} is nonzero if and only if θ(s_{E})_{μ,ν}^{E} is nonzero; so π_{χ} determines an isomorphism (M_{E}^{E,ω})_{χ} ≅ ⨁_{v∈s(E)} M_{Ev}(ℂ). So ψ descends to an isomorphism of each fibre in the trivial bundle ⨁_{v∈s(E)} M_{Ev}(ℂ) ⊗ C^{∗}(A), and so is isometric by [29, Proposition C.10(c)].

Lemma 6.3. Let Γ be a finitely aligned k-graph. Let A be a locally compact abelian group and consider ω ∈ \underline{Z}^{2}(Λ, A). Let (ϕ, π) be a Toeplitz ωrepresentation of (Γ, A). Suppose that E ⊆ F are finite subsets of Γ satisfying E = Ẽ and F = F̃. For each μ ∈ F there is a unique maximal ι_{μ} ∈ E such that μ ∈ ι_{μ}Γ, and we have

Θ(ϕ)_{μ,ν}^{E} = ∑_{μα∈F,ι_{μα}=μ} π(ω(μ, α))π(ω(ν, α))^{∗}Θ(ϕ)_{μα,να}^{F}.

Proof. Let N = ⋁{d(λ) : λ ∈ E, μ ∈ λΓ}. Then N ≤ d(μ) and since E = ∨E, factorising μ = ι_{μ}μ′ with d(ι_{μ}) = N gives the desired ι_{μ}. We claim that for μ ∈ E and λ ∈ F we have Θ(ϕ)_{μ,μ}^{E}Θ(ϕ)_{λ,λ}^{F} = δ_{ι_{λ},μ}Θ(ϕ)_{λ,λ}^{F}. First suppose that ι_{λ} = μ. We have

Θ(ϕ)_{μ,μ}^{E}Θ(ϕ)_{λ,λ}^{F}

= ϕ(μ)ϕ(μ)^{∗} ∏_{μ′∈T (E;μ)} (ϕ(μ)ϕ(μ)^{∗} − ϕ(μμ′)ϕ(μμ′)^{∗})ϕ(λ)ϕ(λ)^{∗}Θ(ϕ)_{λ,λ}^{F}.

For μ′ ∈ T (E; μ) the maximality of ι_{λ} = μ ensures that λ ∉ μμ′Λ. Since F = ∨F and contains E, we deduce that if μμ′α = λβ ∈ MCE(μμ′, λ), then ϕ(λ)ϕ(λ)^{∗}−ϕ(λβ)ϕ(λβ)^{∗} is a factor in Θ(ϕ)_{λ,λ}^{F}, and so ϕ(μμ′)ϕ(μμ′)^{∗}Θ(ϕ)_{λ,λ}^{F} = 0. So the preceding displayed equation collapses to

Θ(ϕ)_{μ,μ}^{E}Θ(ϕ)_{λ,λ}^{F} = ϕ(μ)ϕ(μ)^{∗}Θ(ϕ)_{λ,λ}^{F} = Θ(ϕ)_{λ,λ}^{F}.

Now suppose that μ ≠ ι_{λ}. Since ι_{λ} is the maximal initial segment of λ in E, we have two cases to consider: either ι_{λ} ∈ μΛ \ {μ} or λ ∉ μΛ. First suppose that ι_{λ} ∈ μΛ \ {μ}. Then ϕ(μ)ϕ(μ)^{∗} − ϕ(ι_{λ})ϕ(ι_{λ})^{∗} ≥ Θ(ϕ)_{μ,μ}^{E}. Since (ϕ(μ)ϕ(μ)^{∗} − ϕ(ι_{λ})ϕ(ι_{λ})^{∗}) ⊥ ϕ(λ)ϕ(λ)^{∗} ≥ Θ(ϕ)_{λ,λ}^{F} it follows that Θ(ϕ)_{μ,μ}^{E}Θ(ϕ)_{λ,λ}^{F} = 0. Now suppose that λ ∉ μΛ. Then μα = λβ ∈ MCE(μ, λ) implies β ∈ T (F; λ), and then the argument of the preceding paragraph gives ϕ(μ)ϕ(μ)^{∗}Θ(ϕ)_{λ,λ}^{F} = 0. Hence Θ(ϕ)_{μ,μ}^{E}Θ(ϕ)_{λ,λ}^{F} = 0. This proves the claim. Now fix μ, ν ∈ E. Equations (6.1) and (6.2) imply that

Θ(ϕ)_{μ,ν}^{E} = ∑_{μα∈F} Θ(ϕ)_{μα,μα}^{F}Θ(ϕ)_{μ,μ}^{E}ϕ(μ)ϕ(ν)^{∗},

and the claim reduces this to

∑_{\begin{matrix}μα∈F \\ ι_{μα}=μ \end{matrix}} Θ(ϕ)_{μα,μα}^{F}ϕ(μ)ϕ(ν)^{∗} = ∑_{\begin{matrix}μα∈F \\ ι_{μα}=μ \end{matrix}} Θ(ϕ)_{μα,μα}^{F}ϕ(μα)ϕ(μα)^{∗}ϕ(μ)ϕ(ν)^{∗}.

A by-now familiar computation using the cocycle identity transforms this into

∑_{μα∈F,ι_{μα}=μ} π(ω(μ, α))π(ω(ν, α))^{∗}Θ(ϕ)_{μα,μα}^{F}ϕ(μα)ϕ(να)^{∗},

and another application of (6.1) completes the proof.

Now given Γ, F and ω as above and a finite subset E = Ẽ of Γ, we define a map ι^{E} : Γ → Γ as follows: if λ ∈ EΓ, then ι_{λ}^{E} is the maximal element of E such that λ ∈ ι_{λ}^{E}Γ. For all other λ ∈ Γ we set ι_{λ}^{E} = λ. We define τ^{E} : Γ → Γ by λ = ι_{λ}τ_{λ} for all λ.

Theorem 6.4. Let Γ be a finitely aligned k-graph. Suppose that there is a map b : Γ^{0} → ℤ^{k} such that d(λ) = b(s(λ)) − b(r(λ)) for all λ and that F ⊆ FE(Γ) is satiated. Let A be a locally compact abelian group and consider ω ∈ \underline{Z}^{2}(Λ, A). There is an isomorphism C^{∗}(Λ, A, ω; F) ≅ C^{∗}(Λ; F) ⊗ C^{∗}(A) which carries ι_{Γ}^{F,ω}(λ)ι_{A}^{F,ω}(f)ι_{Γ}^{F,ω}(λ)^{∗} to s_{F}(λ)s_{F}(λ)^{∗} ⊗ f for all λ ∈ Λ.

Proof. Fix an increasing sequence E_{1} ⊆ E_{2} ⊆ E_{3} ⊆ ⋯ of subsets of Γ such that each E_{i} = Ẽ_{i} and ⋃_{i} E_{i} = Γ. Recursively define maps κ_{i} : {(μ, ν) ∈ Γ : s(μ) = s(ν)} → MC^{∗}(A) by κ_{0} ≡ 1_{MC^{∗}(A)} and

κ_{i+1}(μ, ν) = κ_{i}(μ, ν) − ω(ι_{μ}^{E_{i}}, τ_{μ}^{E_{i}}) + ω(ι_{ν}^{E_{i}}, τ_{ν}^{E_{i}}).

For each i, define a linear map ψ_{i} : M_{E_{i}}^{F,ω} → M_{E_{i}}^{F} by

ψ_{i}(ι_{A}^{F,ω}(f)Θ(ι_{Γ}^{F,ω})_{μ,ν}^{E_{i}}) = ι_{A}^{F,ω}(κ_{i}(μ, ν))ι_{A}^{F,ω}(f)Θ(ι_{Γ}^{F})_{μ,ν}^{E_{i}}.

An induction argument shows that each κ_{i} satisfies κ_{i}(ν, μ) = −κ_{i}(μ, ν) and that κ_{i}(λ, μ) + κ_{i}(μ, ν) = κ_{i}(λ, ν), and so ψ_{i} preserves adjoints and multiplication. Two applications of Lemma 6.2 show that ψ_{i} is an isomorphism. Using the definition of the κ_{i} and Lemma 6.3, we calculate:

ψ_{i}(Θ(ι_{Γ}^{F,ω})_{μ,ν})

= ι_{A}^{F,ω}(κ_{i}(μ, ν))Θ(ι_{Γ}^{F})_{μ,ν}^{E_{i}}

= ∑_{μα∈E_{i+1},ι_{μα}=μ} ι_{A}^{F,ω}(κ_{i}(μ, ν))Θ(ι_{Γ}^{F})_{μα,να}^{E_{i+1}}

= ∑_{μα∈E_{i+1},ι_{μα}=μ} ι_{A}^{F,ω}(κ_{i+1}(μα, να) + ω(μ, α) − ω(ν, α))Θ(ι_{Γ}^{F})_{μα,να}^{E_{i+1}}

= ∑_{μα∈E_{i+1},ι_{μα}=μ} ψ_{i+1}(ι_{A}^{F,ω}(ω(μ, α))ι_{A}^{F,ω}(ω(ν, α))^{∗}Θ(ι_{Γ}^{F,ω})_{μα,να}^{E_{i+1}})

= ψ_{i+1}(Θ(ι_{Γ}^{F,ω})_{μ,ν}^{E_{i}}).

Hence there is an isomorphism ψ_{∞} : C^{∗}(Γ, A, ω; F) → C^{∗}(Γ, A; F) such that ψ_{∞}|_{M_{E_{i}}^{F,ω}} = ψ_{i}. The formula (6.2) gives ψ_{∞}(ι_{Γ}^{F,ω}(λ)ι_{A}^{F,ω}(f)ι_{Γ}^{F,ω}(λ)^{∗}) =

s_{F}(λ)s_{F}(λ)^{∗} ⊗ f for each λ and f.

If A is a C_{0}(X)-algebra in the sense that there is a nondegenerate homomorphism i : C_{0}(X) → ZM(A), and if I ⊆ X is closed, then we write A|_{I} for the quotient of A by the ideal generated by {i(f) : f|_{I} = 0}.

Corollary 6.5. Let Λ be a finitely aligned k-graph and suppose that E ⊆ FE(Λ) is satiated. Suppose that ω ∈ \underline{Z}^{2}(Λ, ℝ). There is an isomorphism C^{∗}(Λ ×_{d} ℤ^{k}, A, ω × 1; E × ℤ^{k})|_{[0,1]} ≅ C^{∗}(Λ ×_{d} ℤ^{k}; E × ℤ^{k}) ⊗ C([0, 1]) that carries

ι_{Λ×_{d}ℤ^{k}}^{E×ℤ^{k}, ω×1}(λ)ι_{A}^{E×ℤ^{k}, ω×1}(f)ι_{Λ×_{d}ℤ^{k}}^{E×ℤ^{k}, ω×1}(λ)^{∗} to s_{E×ℤ^{k}}^{χ∘ω}(λ)s_{E×ℤ^{k}}^{χ∘ω}(λ)^{∗} ⊗ f for all λ, f. In particular, evaluation at any character χ ∈ [0, 1] ⊆ ℝ̂ determines an isomorphism K_{∗}(C^{∗}(Λ ×_{d} ℤ^{k}, A, ω × 1; E × ℤ^{k})|_{[0,1]}) ≅ K_{∗}(C^{∗}(Λ ×_{d} ℤ^{k}, χ ∘ ω; E × ℤ^{k}))

which carries [ι_{Λ×_{d}ℤ^{k}}^{E×ℤ^{k}, ω×1}(λ)ι_{Λ×_{d}ℤ^{k}}^{E×ℤ^{k}, ω×1}(λ)^{∗}] to [s_{E×ℤ^{k}}^{χ∘ω}(λ)s_{E×ℤ^{k}}^{χ∘ω}(λ)^{∗}].

Proof. Use Theorem 6.4 and the Künneth theorem (see [17, Corollary 4.3]).

We can now prove the main result of the section, Theorem 6.1

Proof of Theorem 6.1. We follow the proof of [17, Theorem 5.4]. As in Corollary 5.2, the argument of [25, Lemma 8.3] shows that C^{∗}(Λ ×_{d} ℤ^{k}, ℝ, ω × 1; E × ℤ^{k}) is isomorphic to the crossed product of C^{∗}(Λ, ℝ, ω; E) by the gauge action and that the inclusion of C^{∗}(Λ, ℝ, ω; E) into the crossed product takes the K_{0} class of an ι_{Λ}^{E,ω}(v) to the class of ι_{Λ×_{d}ℤ^{k}}^{E×ℤ^{k};ω×1}((v, 0)). The argument of [17, Lemma 5.2] goes through more or less verbatim (substitute “relative Cuntz-Krieger family” for “Cuntz-Krieger family” as necessary) to prove that translation in the ℤ^{k} coordinate on Λ ×_{d} ℤ^{k} induces the action γ̂ of ℤ^{k} on C^{∗}(Λ ×_{d} ℤ^{k}, ℝ, ω; E) that is dual to the gauge-action, that the projection P_{0} = ∑_{v∈Λ^{0}} ι_{Λ×_{d}ℤ^{k}}^{E×ℤ^{k};ω×1}((v, 0)) is full in C^{∗}(Λ ×_{d} ℤ^{k}, ℝ, ω × 1; E × ℤ^{k}) ×γ̂ ℤ^{k} and that the corner it determines is isomorphic to C^{∗}(Λ, ℝ, ω; E) via an isomorphism that takes the generating projection associated to (v, 0) ∈ (Λ ×_{d} ℤ^{k})^{0} to the generating projection associated to v ∈ Λ. Lemma 5.3 of [17] implies that the inclusion of C^{∗}(ℝ) in the centre of C^{∗}(Λ ×_{d} ℤ^{k}, ℝ, ω × 1; E × ℤ^{k}) ×γ̂ ℤ^{k} makes it into a C_{0}(ℝ) algebra whose fibre over χ ∈ ℝ̂ is C^{∗}(Λ ×_{d} ℤ^{k}, χ ∘ (ω × 1); E × ℤ^{k}) ×_{}_{γ̂^{χ∘ω}} ℤ^{k}. Since Corollary 6.5 implies that evaluation at each point of [0, 1] induces an isomorphism in K-theory on C^{∗}(Λ ×_{d} ℤ^{k}, ℝ, ω × 1; E × ℤ^{k})|_{[0,1]}, Theorem 5.1 of [17] implies that the same is true of the crossed product; applying this at t = 0 and t = 1 gives the result.

7 Simplicity

In Corollary 3.18 we showed that if Λ is cofinal and aperiodic in the sense of [18], then each C^{∗}(Λ, c) is simple. In the untwisted setting, these conditions are also necessary (see [18, Theorem 3.4]), but the example of rotation algebras, discussed in the final paragraph of the introduction, shows that this is not to be expected in the twisted setting. In this section we give a sufficient condition for simplicity of twisted C^{∗}-algebras associated to k-graphs that are not aperiodic. A bicharacter of ℤ^{k} is a map c : ℤ^{k} × ℤ^{k} → 𝕋 such that

c(m, n)c(m, n′) = c(m, n + n′)  and  c(m, n)c(m′, n) = c(m + m′, n).

This implies that c(0, n) = c(m, 0) = 1 and that c(−m, n) = \overline{c(m, n)} = c(m, −n) for all m, n. A bicharacter is skew-symmetric^{1} if it has the additional property that c(n, m) = \overline{c(m, n)} for all m, n. For c ∈ Z^{2}(ℤ^{k}, 𝕋), we write c^{∗} for the cocycle c^{∗}(m, n) = \overline{c(n, m)} (note the reversal of variables; so c^{∗} ≠ \overline{c}). Proposition 3.2 of [19] shows that the product cc^{∗}, given by (cc^{∗})(m, n) = c(m, n)\overline{c(n, m)}, is a skew-symmetric bicharacter, and moreover that the map c ↦ cc^{∗} has kernel B^{2}(ℤ^{k}, 𝕋) and determines an isomorphism of H^{2}(ℤ^{k}, 𝕋) onto the group of skew-symmetric bicharacters of ℤ^{k}. It follows immediately (or see [2]) that H^{2}(ℤ^{k}, 𝕋) is isomorphic to 𝕋^{k(k−1)/2}, and that every class has a representative that is a bicharacter: given c ∈ Z^{2}(ℤ^{k}, 𝕋), form the skew-symmetric bicharacter cc^{∗}, and let c̃ be the unique bicharacter of ℤ^{k} such that c̃(e_{i}, e_{j}) = cc^{∗}(e_{i}, e_{j}) if i > j and c̃(e_{i}, e_{j}) = 1 if i ≤ j. Then c̃c̃^{∗}(e_{i}, e_{j}) = cc^{∗}(e_{i}, e_{j}) for all i, j, and since both are bicharacters, the two agree. Hence [19, Proposition 3.2] discussed above shows that c and c̃ represent the same class in H^{2}(ℤ^{k}, 𝕋). A skew-symmetric bicharacter c is called nondegenerate if there is no nonzero m ∈ ℤ^{k} such that c(m, n) = 1 for all n. So if c ∈ Z^{2}(ℤ^{k}, 𝕋) is a bicharacter, then cc^{∗} is nondegenerate if and only if there is no nonzero m such that c(n, m) = c(m, n) for all n (equivalently, no nonzero m satisfies c(m, e_{i}) = c(e_{i}, m) for all i ∈ {1, ⋯, k}). Given c ∈ Z^{2}(ℤ^{k}, 𝕋), the noncommutative torus A(c) is the universal algebra generated by unitaries {U_{m} : m ∈ ℤ^{k}} satisfying U_{m}U_{n} = c(m, n)U_{m+n} for all m, n ∈ ℤ^{k}. An argument like that of Proposition 3.5 shows that if c_{1} and c_{2} are cohomologous in Z^{2}(ℤ^{k}, 𝕋), then A(c_{1}) ≅ A(c_{2}). So the remarks in the preceding paragraph (see also Remark 1.2 of [20]) show that A(c) is universal for unitaries {U_{i} : i ≤ k} satisfying U_{i}U_{j} = cc^{∗}(e_{i}, e_{j})U_{j}U_{i}. Theorem 3.7 of [27] combined with 1.8 of [7] (see [20, Theorem 1.9]) shows that the noncommutative torus A(c) is simple if and only if cc^{∗} is nondegenerate. To state the main result of the section, observe that if Λ is row-finite and has no sources, then vΛ^{n} ∈ FE(Λ) for each v ∈ Λ^{0} and n ∈ ℕ^{k} \ {0}. Hence a filter S is FE(Λ)-compatible if and only if it is maximal in the sense that S ∩ Λ^{n} ≠ ∅ for all n ∈ ℕ^{k}. We call such filters ultrafilters. Recall that if S is an ultrafilter, and s(μ) = r(S), then ℓ_{μ}(S) is the filter {ν ∈ λ : νΛ ∩ μS ≠ ∅}, which is then also an ultrafilter. Under the bijection of Remark 3.8, the ultrafilters of Λ correspond to the infinite paths used in [12], and if this bijection carries the infinite path x to the ultrafilter S, then for μ ∈ Λr(x), it carries the infinite path μx to the ultrafilter ℓ_{μ}(S). As in [4], for μ, ν in a row-finite k-graph Λ with no sources, we write μ ∼ ν if μx = νx for every infinite path x in s(μ)Λ^{∞}. Equivalently, μ ∼ ν if ℓ_{μ}(S) = ℓ_{ν}(S) for every ultrafilter S such that r(S) = s(μ). We define Per(Λ) to be the subgroup of ℤ^{k} generated by {d(μ) − d(ν) : μ ∼ ν}. In this section, if c ∈ Z^{2}(ℤ^{k}, 𝕋) and Λ is a k-graph with degree functor d, then we abuse notation slightly and write c ∘ d for the cocycle c ∘ d(λ, μ) := c(d(λ), d(μ)).

1symplectic in the language of [19].

Documenta Mathematica 19 (2014) 831–866

Theorem 7.1. Let Λ be a row-finite k-graph with no sources, and take c ∈ Z^{2}(ℤ^{k}, 𝕋). Suppose that (cc^{∗})|_{Per(Λ)} is nondegenerate. Then C^{∗}(Λ, c ∘ d) is simple if and only if Λ is cofinal.

For our first couple of results, we continue to work in the generality of finitely aligned k-graphs. We begin by showing that cofinality of Λ is necessary for simplicity of C^{∗}(Λ, c).

Lemma 7.2. Let Λ be a finitely aligned k-graph, and suppose that c ∈ \underline{Z}^{2}(Λ, 𝕋). Suppose that Λ is not cofinal. Then C^{∗}(Λ, c) is not simple.

Proof. By [18, Theorem 5.1] there exists v ∈ Λ^{0} and an FE(Λ)-compatible filter S such that vΛs(λ) = ∅ for all λ ∈ S (that is, vΛs(S) = ∅). Consider

H_{S} := \overline{span}\{h_{μ} : s(μ) ∈ s(S)\} ⊆ ℓ^{2}(Λ).

Let {T_{λ} : λ ∈ Λ} be the twisted Toeplitz-Cuntz-Krieger family on ℓ^{2}(Λ) described just before Proposition 3.9, and let π_{T} be the associated representation of T C^{∗}(Λ). For μ, ν ∈ Λ and a basis element h_{η} of H_{S} we have

T_{μ}T_{ν}^{∗}h_{η} = \begin{cases}c(μ, η′)\overline{c(ν, η′)}h_{μη′} & if η = νη′ \\ 0 & otherwise. \end{cases}

Since s(μη′) = s(η) ∈ s(S), it follows that H_{S} is an invariant subspace of ℓ^{2}(Λ) for π_{T}. Hence restriction determines a representation π_{T}^{H_{S}} of T C^{∗}(Λ, c) on H_{S}. We have T_{v}h_{η} = 0 for all η ∈ Λs(S) because vΛs(S) = ∅. Thus s_{T}^{c} (v) ∈ ker(π_{T}^{H_{S}}). Let J = J_{FE(Λ)} be the ideal of T C^{∗}(Λ, c) generated by {Δ(s_{T}^{c})^{E} : E ∈ FE(Λ)}, so that T C^{∗}(Λ, c)/J = C^{∗}(Λ, c). Proposition 3.10 implies that lim_{μ∈S} π_{T} (a)h_{μ} = 0 for all a ∈ J. Since ∥π_{T} (s_{T}^{c} (r(S)))h_{μ}∥ = ∥h_{μ}∥ = 1 for all μ ∈ S, we deduce that

∥π_{T} (s_{T}^{c} (r(S)) − a)|_{H_{S}} ∥ ≥ 1 for all a ∈ J.  (7.1)

Let I := π_{T}^{H_{S}}(J). Then I is an ideal of π_{T}^{H_{S}}(T C^{∗}(Λ, c)). Let q_{I} be the quotient map from π_{T}^{H_{S}}(T C^{∗}(Λ, c)) to π_{T}^{H_{S}}(T C^{∗}(Λ, c))/I. Equation (7.1) implies that ∥q_{I} ∘ π_{T}^{H_{S}}(s_{T}^{c} (r(S)))∥ = 1, and so q_{I} ∘ π_{T}^{H_{S}} induces a nonzero homomorphism ρ of C^{∗}(Λ, c). The preceding paragraph shows that q_{I} ∘ π_{T}^{H_{S}}(s_{T}^{c} (v)) = 0. Hence ρ is neither faithful nor trivial, and it follows that C^{∗}(Λ, c) is not simple.

Proposition 7.3. Let Λ be a finitely aligned k-graph, and let c ∈ \underline{Z}^{2}(Λ, 𝕋). There is a faithful conditional expectation

Θ : C^{∗}(Λ, c) → D_{Λ}^{c} := \overline{span}\{s^{c}(λ)s^{c}(λ)^{∗} : λ ∈ Λ\}

given by Θ(s^{c}(μ)s^{c}(ν)^{∗}) = δ_{μ,ν}s^{c}(μ)s^{c}(μ)^{∗}.

Proof. Averaging over the gauge action γ gives a faithful conditional expectation Φ : C^{∗}(Λ, c) → C^{∗}(Λ, c)^{γ} = \overline{span}\{s^{c}(μ)s^{c}(ν)^{∗} : d(μ) = d(ν)\} such that Φ(s^{c}(μ)s^{c}(ν)^{∗}) = δ_{d(μ),d(ν)}s^{c}(μ)s^{c}(ν)^{∗}. Lemma 3.12 shows that C^{∗}(Λ, c)^{γ} can be written as the closure of an increasing union of finite-dimensional subalgebras M(s^{c})_{E} with nested diagonal subalgebras whose union is dense in D_{Λ}^{c}. Equation (3.2) shows that if μ ≠ ν then s^{c}(μ)s^{c}(ν)^{∗} belongs to the span of the off-diagonal matrix units in M(s^{c})_{E} for sufficiently large E. So the canonical faithful expectation Ψ from the AF algebra C^{∗}(Λ, c)^{γ} onto its diagonal subalgebra satisfies Θ(s^{c}(μ)s^{c}(ν)^{∗}) = δ_{μ,ν}s^{c}(μ)s^{c}(μ)^{∗}. Now Θ = Ψ ∘ Φ is the desired expectation.

We now restrict our attention to row-finite k-graphs with no sources. Recall that in this setting the Cuntz-Krieger relations for C^{∗}(Λ, c) imply that

∑_{λ∈vΛ^{n}} s^{c}(λ)s^{c}(λ)^{∗} = s^{c}(v) for every v ∈ Λ^{0} and n ∈ ℕ^{k}.  (7.2)

It follows from this and (TCK2) that

s^{c}(λ)s^{c}(λ)^{∗} = ∑_{μ∈λΛ^{n}} s^{c}(μ)s^{c}(μ)^{∗} for all λ ∈ Λ and n ∈ ℕ^{k}.  (7.3)

Lemma 7.4. Let Λ be a row-finite k-graph with no sources. Suppose that μ ∼ ν in Λ. Then MCE(μ, ν) = μΛ ∩ Λ^{d(μ)∨d(ν)} = νΛ ∩ Λ^{d(μ)∨d(ν)}.

Proof. For any μα ∈ μΛ ∩ Λ^{d(μ)∨d(ν)} and any ultrafilter S with s(α) ∈ S, we have ℓ_{μα}(S) = ℓ_{μ}(ℓ_{α}(S)) = ℓ_{ν}(ℓ_{α}(S)) = ℓ_{να}(S), and so μα ∈ νΛ ∩ Λ^{d(μ)∨d(ν)}. So μΛ∩Λ^{d(μ)∨d(ν)} ⊆ νΛ∩Λ^{d(μ)∨d(ν)}, and symmetry gives the reverse inclusion. Since MCE(μ, ν) = μΛ ∩ νΛ ∩ Λ^{d(μ)∨d(ν)}, the result follows.

Recall from [4, Theorem 4.2] that if Λ is cofinal, then Per(Λ) = {d(μ) − d(ν) : μ ∼ ν}. Since Per(Λ) it is a subgroup of ℤ^{k}, it is isomorphic to ℤ^{l} for some l ≤ k. The collection

H_{Per} = {v ∈ Λ^{0} : whenever m, n ∈ ℕ^{k}, m − n ∈ Per(Λ) and λ ∈ vΛ^{m} there
exists μ ∈ vΛ^{n} such that λ ∼ μ}

is a nonempty hereditary subset of Λ^{0}. For m, n ∈ ℕ^{k} such that m−n ∈ Per(Λ), there is a bijection θ_{m,n} : H_{Per}Λ^{m} → H_{Per}Λ^{n} such that μ ∼ θ_{m,n}(μ) for all μ; this θ_{m,n} preserves the range and source maps. Suppose that μ ∼ ν ∈ Λ, and write m = d(μ) and n = d(ν). Let t be a Cuntz-Krieger (Λ, c)-family. Then (7.3) implies that t_{μ}t_{μ}^{∗} = ∑_{η∈μΛ^{(m∨n)−m}} t_{η}t_{η}^{∗}, and then Lemma 7.4 implies that t_{μ}t_{μ}^{∗} = t_{μ}t_{μ}^{∗}t_{ν}t_{ν}^{∗}; that is, t_{μ}t_{μ}^{∗} ≤ t_{ν}t_{ν}^{∗}. Symmetry gives the reverse inequality, and hence

t_{μ}t_{μ}^{∗} = t_{ν}t_{ν}^{∗}  whenever μ ∼ ν.  (7.4)

For the following proposition, we remind the reader that A(c) denotes the noncommutative torus, described at the beginning of this section.

Proposition 7.5. Let Λ be a row-finite k-graph with no sources. Let c ∈ Z^{2}(ℤ^{k}, 𝕋), and suppose that (cc^{∗})|_{Per(Λ)} is nondegenerate. Suppose that t is a Cuntz-Krieger (Λ, c ∘ d)-family such that each t_{v} ≠ 0. Suppose that F is a finite subset of H_{Per}Λ^{n}, and let F̃ := {ν ∈ Λ : ν ∼ μ for some μ ∈ F }. There is an injective homomorphism ⨁_{λ∈F} ρ_{t}^{λ} : ⨁_{λ∈F} A(c|_{Per(Λ)}) → C^{∗}(t) such that ρ_{t}^{λ}(U_{d(λ)−d(ν)}) = t_{λ}t_{ν}^{∗} whenever λ ∈ F and ν ∼ λ.

Proof. As discussed at the beginning of the section, we may assume that c is a bicharacter. If μ, ν ∈ F̃ and μ ≁ ν, then there exist distinct λ, λ′ ∈ F such that μ ∼ λ and ν ∼ λ′. Equation (7.4) gives t_{μ}t_{μ}^{∗} = t_{λ}t_{λ}^{∗} ⊥ t_{λ′}t_{λ′}^{∗} = t_{ν}t_{ν}^{∗}. Hence

C^{∗}({t_{μ}t_{ν}^{∗} : μ, ν ∈ F̃ and μ ∼ ν}) = ⨁_{λ∈F} C^{∗}({t_{μ}t_{ν}^{∗} : μ ∼ λ ∼ ν}).

Since each t_{v} is nonzero, so is each t_{μ}t_{ν}^{∗}, and so each summand C^{∗}({t_{μ}t_{ν}^{∗} : μ ∼ λ ∼ ν}) is nontrivial. Fix λ ∈ F. It now suffices to show that there is an injective homomorphism ρ_{s} : A(c|_{Per(Λ)}) → t_{λ}t_{λ}^{∗}C^{∗}(t)t_{λ}t_{λ}^{∗} such that ρ_{s}(U_{d(μ)−d(ν)}) = t_{μ}t_{ν}^{∗} whenever μ ∼ λ ∼ ν. To see this, for each m ∈ Per(Λ), express m = p(m) − q(m) where p(m), q(m) ∈ ℕ^{k} and p(m) = n whenever m ≤ n. Since r(λ) ∈ H_{Per}, we can define

V_{m} := ∑_{μ∈r(λ)Λ^{p(m)}} t_{λ}t_{λ}^{∗}t_{μ}t_{θ_{p(m),q(m)}(μ)}^{∗}.  (7.5)

We have

V_{m}V_{m}^{∗} = ∑_{μ,ν∈r(λ)Λ^{p(m)}} t_{λ}t_{λ}^{∗}t_{μ}t_{θ_{p(m),q(m)}(μ)}^{∗}t_{θ_{p(m),q(m)}(ν)}t_{ν}^{∗}t_{λ}t_{λ}^{∗}.

Each d(θ_{p(m),q(m)}(μ)) = q(m) = d(θ_{p(m),q(m)}(ν)) and θ_{p(m),q(m)} is a bijection, so the Cuntz-Krieger relation (7.2) gives

V_{m}V_{m}^{∗} = ∑_{μ∈r(λ)Λ^{p}} t_{λ}t_{λ}^{∗}t_{μ}t_{μ}^{∗}t_{λ}t_{λ}^{∗},

and then (7.2) again reduces this to t_{λ}t_{λ}^{∗}. To see that V_{m}^{∗}V_{m} is also equal to t_{λ}t_{λ}^{∗}, observe that

V_{m}^{∗}V_{m} = ∑_{μ,ν∈r(λ)Λ^{p(m)}} t_{θ_{p(m),q(m)}(μ)}t_{μ}^{∗}t_{λ}t_{λ}^{∗}t_{ν}t_{θ_{p(m),q(m)}(ν)}^{∗}

= ∑_{\begin{matrix}μ,ν∈r(λ)Λ^{p(m)} \\ η∈λΛ^{p(m)} \end{matrix}} t_{θ_{p(m),q(m)}(μ)}t_{μ}^{∗}t_{η}t_{η}^{∗}t_{ν}t_{θ_{p(m),q(m)}(ν)}^{∗}.

For each η ∈ λΛ^{p(m)}, factorise η = α_{η}β_{η} with d(α_{η}) = p(m) and d(β_{η}) = n. For μ ∈ r(λ)Λ^{p(m)} and η ∈ λΛ^{p(m)} we have d(μ) = p(m) = d(α_{η}), so the first displayed equation in Lemma 3.2 implies that t_{μ}^{∗}t_{α_{η}} = δ_{μ,α_{η}}t_{s(μ)}. Thus

V_{m}^{∗}V_{m} = ∑_{η∈λΛ^{p(m)}} t_{θ_{p(m),q(m)}(α_{η})}t_{β_{η}}t_{β_{η}}^{∗}t_{θ_{p(m),q(m)}(α_{η})}^{∗}.

If μ ∼ ν then the factorisation property gives μτ ∼ ντ for all τ, and so θ_{m+p,n+p}(μτ) = θ_{m,n}(μ)τ whenever μ ∈ Λ^{m} and τ ∈ s(μ)Λ^{p}. This and (7.4) give

\begin{aligned}V_{m}^{∗}V_{m} & = ∑_{η∈λΛ^{p(m)}} t_{θ_{p(m)+n,q(m)+n}(η)}t_{θ_{p(m)+n,q(m)+n}(η)}^{∗} \\ & = ∑_{η∈λΛ^{p(m)}} t_{η}t_{η}^{∗} = t_{λ}t_{λ}^{∗}. \end{aligned}  (7.6)

Hence the V_{m} are unitaries in t_{λ}t_{λ}^{∗}C^{∗}(t)t_{λ}t_{λ}^{∗}. Combining (7.5) and (7.6) we now have

∑_{μ∈r(λ)Λ^{p(m)}} t_{λ}t_{λ}^{∗}t_{μ}t_{θ_{p(m),q(m)}(μ)}^{∗} = V_{m} = V_{m}t_{λ}t_{λ}^{∗}

= ∑_{μ∈r(λ)Λ^{p(m)}} t_{λ}t_{λ}^{∗}t_{μ}t_{θ_{p(m),q(m)}(μ)}^{∗}t_{λ}t_{λ}^{∗},

and then symmetry gives

V_{m} = ∑_{μ∈r(λ)Λ^{p(m)}} t_{μ}t_{θ_{p(m),q(m)}(μ)}^{∗}t_{λ}t_{λ}^{∗}.

Fix m, m′ in Per(Λ), and write p := p(m), p′ := p(m′), q := q(m) and q′ := q(m′). For the next few calculations, put

C_{m,m′} := c(p, p′)\overline{c(q, p′)}c(p′, q)\overline{c(q′, q)}.

Then relation (7.2) gives

V_{m}V_{m′} = ∑_{μ∈r(λ)Λ^{p}} t_{λ}t_{λ}^{∗}t_{μ}t_{θ_{p,q}(μ)}^{∗} ∑_{η∈r(λ)Λ^{p′}} t_{η}t_{θ_{p′,q′} (η)}^{∗}t_{λ}t_{λ}^{∗}

= ∑_{\begin{matrix}μ∈r(λ)Λ^{p}, μ′∈s(μ)Λ^{p′} \\ η∈r(λ)Λ^{p′} , η′∈s(η)Λ^{q} \end{matrix}} C_{m,m′}t_{λ}t_{λ}^{∗}t_{μμ′}t_{θ_{p,q}(μ)μ′}^{∗}t_{ηη′}t_{θ_{p′,q′} (η)η′}^{∗}t_{λ}t_{λ}^{∗}

= C_{m,m′} ∑_{\begin{matrix}μ∈r(λ)Λ^{p}, μ′∈s(μ)Λ^{p′} \\ η∈r(λ)Λ^{p′} , η′∈s(η)Λ^{q} \end{matrix}} δ_{θ_{p,q}(μ)μ′,ηη′}t_{λ}t_{λ}^{∗}t_{μμ′}t_{θ_{p′,q′} (η)η′}^{∗}t_{λ}t_{λ}^{∗}.

If θ_{p,q}(μ)μ′ = ηη′, then μμ′ ∼ θ_{p,q}(μ)μ′ = ηη′ ∼ θ_{p′,q′}(η)η′; and conversely μμ′ ∼ θ_{p′,q′}(η)η′ implies θ_{p,q}(μ)μ′ = ηη′ by a symmetric argument. So the final line of the preceding displayed equation becomes

C_{m,m′} ∑_{\begin{matrix}μ∈r(λ)Λ^{p}, μ′∈s(μ)Λ^{p′} \\ ζ∈r(λ)Λ^{q′} , ζ′∈s(η)Λ^{q} \\ μμ′∼ζζ′ \end{matrix}} t_{λ}t_{λ}^{∗}t_{μμ′}t_{ζζ′}^{∗}t_{λ}t_{λ}^{∗}.  (7.7)

Let a := p(m + m′) and b := q(m + m′). Then there are h, l ∈ ℕ^{k} such that p + p′ + h = a + l and q + q′ + h = b + l. For α ∈ Λ^{p+p′} and β ∈ Λ^{q+q′}, we have α ∼ β if and only if there is an injection τ ↦ (η(τ), ρ(τ), ζ(τ)) from s(α)Λ^{h} to r(α)Λ^{a} × Λ^{l}s(α) × r(α)Λ^{b} such that ατ = η(τ)ρ(τ) and βτ = ζ(τ)ρ(τ) and η(τ) ∼ ζ(τ) for each τ. Using this, and applying relation (7.2), we obtain

∑_{\begin{matrix}μ∈r(λ)Λ^{p}, μ′∈s(μ)Λ^{p′} \\ ζ∈r(λ)Λ^{q′} , ζ′∈s(η)Λ^{q} \\ μμ′∼ζζ′, \end{matrix}}t_{μμ′}t_{ζζ′}^{∗}

= ∑_{\begin{matrix}μ∈r(λ)Λ^{p}, μ′∈s(μ)Λ^{p′} \\ ζ∈r(λ)Λ^{q′} , ζ′∈s(η)Λ^{q} \\ μμ′∼ζζ′, τ∈s(μ′)Λ^{h} \end{matrix}} c(p + p′, h)\overline{c(q + q′, h)}t_{μμ′τ} t_{ζζ′τ}^{∗}

= ∑_{\begin{matrix}μ∈r(λ)Λ^{p}, μ′∈s(μ)Λ^{p′} \\ ζ∈r(λ)Λ^{q′} , ζ′∈s(η)Λ^{q} \\ μμ′∼ζζ′, τ∈s(μ′)Λ^{h} \end{matrix}} c(m + m′, h)t_{μμ′τ} t_{ζζ′τ}^{∗} .

Hence the expression (7.7) for V_{m}V_{m}^{∗} gives

V_{m}V_{m′} = C_{m,m′}c(m + m′, h) ∑_{η∈r(λ)Λ^{a}, ρ∈s(η)Λ^{l}} t_{λ}t_{λ}^{∗}t_{ηρ}t_{θ_{a,b}(η)ρ}^{∗}t_{λ}t_{λ}^{∗}
= C_{m,m′}c(m + m′, h)\overline{c(a, l)}c(b, l)

∑_{η∈r(λ)Λ^{a}, ρ∈s(η)Λ^{l}} t_{λ}t_{λ}^{∗}t_{α′}t_{τ} t_{τ}^{∗}t_{θ_{a,b}(α′)}^{∗}t_{λ}t_{λ}^{∗}

= c(p, p′)\overline{c(q, p′)}c(p′, q)\overline{c(q′, q)}c(m + m′, h − l)V_{m+m′}.

Rearranging this expression for V_{m}V_{m′} and the symmetric expression

V_{m′}V_{m} = c(p′, p)\overline{c(q′, p)}c(p, q′)\overline{c(q, q′)}c(m + m′, h − l)V_{m+m′},

and cancelling the occurrences of c(m + m′, h − l), we obtain

\begin{matrix}V_{m}V_{m′} = c(p, p′) & \overline{c(q, p′)}\overline{c(p, q′)}c(q, q′) \\ \overline{c(p′, p)}c(p′, q)\overline{c(q′, q)}c(q′, p)V_{m′}V_{m}. \end{matrix}  (7.8)

Using repeatedly that c is a bicharacter, we calculate:

c(p, p′)\overline{c(q, p′)}\overline{c(p, q′)}c(q, q′)\overline{c(p′, p)}c(p′, q)\overline{c(q′, q)}c(q′, p)
= c(p − q, p′)c(−p + q, q′)c(p′, −p + q)c(q′, p − q)
= c(p − q, p′)c(p − q, −q′)\overline{c(p′, p − q)c(−q′, p − q)}
= c(m, m′)\overline{c(m′, m)}.

Documenta Mathematica 19 (2014) 831–866

Thus (7.8) becomes

V_{m}V_{m′} = c(m, m′)\overline{c(m′, m)}V_{m′}V_{m}.

Choose generators g_{1}, …, g_{l} for Per(Λ). Then each V_{m} is a scalar multiple of a product of the V_{g_{i}}, and so C^{∗}({V_{m} : m ∈ Per(Λ)}) is generated by the V_{g_{i}}, which satisfy V_{g_{i}}V_{g_{j}} = (cc^{∗})(g_{i}, g_{j})V_{g_{j}}V_{g_{i}}. Since c_{Per} := c|_{Per(Λ)} is nondegenerate, [27, Theorem 3.7] implies that there is an isomorphism ρ_{t}^{λ} : A(c_{Per}) → C^{∗}({V_{m} : m ∈ Per(Λ)}) carrying each U_{m} to V_{m}. Recall that we chose p(m) = n whenever m ≤ n. So if λ ∼ ν then

V_{n−d(ν)} = t_{λ}t_{λ}^{∗} ∑_{μ∈r(λ)Λ^{n}} t_{μ}t_{θ_{n,d(ν)}(μ)}^{∗} = t_{λ}t_{θ_{n,d(ν)}(λ)}^{∗} = t_{λ}t_{ν}^{∗}.

So ρ_{t}^{λ} carries U_{d(λ)−d(ν)} to t_{λ}t_{ν}^{∗} whenever ν ∼ λ. Hence ⨁_{λ∈F} ρ_{t}^{λ} : ⨁_{λ∈F} A(c_{Per}) → C^{∗}(t) is the desired homomorphism.

Lemma 7.6. Let Λ be a row-finite k-graph with no sources. Suppose that μ, ν ∈ Λ satisfy s(μ) = s(ν). Then μ ∼ ν if and only if MCE(μτ, ντ) ≠ ∅ for all τ ∈ s(μ)Λ.

Proof. First suppose that μ ∼ ν, and fix τ ∈ s(μ)Λ. There is an ultrafilter S with τ ∈ S, and we have ℓ_{μ}(S) = ℓ_{ν}(S). Since S is an ultrafilter, so is ℓ_{μ}(S), and so ℓ_{μ}(S)∩Λ^{d(μτ)∨d(ντ)} ≠ ∅, and then the unique element of ℓ_{μ}(S)∩Λ^{d(μτ)∨d(ντ)} belongs to MCE(μτ, ντ). Now suppose that MCE(μτ, ντ) ≠ ∅ for all τ. Using the correspondence between ultrafilters and infinite paths of Remark 3.8 together with the topology on the space of infinite paths described in [12, Definition 2.4 and Lemma 2.6], we see that the sets Z(λ) = {S : S is an ultrafilter and λ ∈ S} are a basis of compact open sets for a Hausdorff topology on the set of all ultrafilters. Fix an ultrafilter S with s(μ) ∈ S. For each τ ∈ S, we have MCE(μτ, ντ) ≠ ∅, and hence Z(μτ) ∩ Z(ντ) ≠ ∅. Since the Z(μτ) ∩ Z(ντ) are compact and decreasing with respect to the partial ordering ≤ on S, we deduce that ⋂_{τ∈S} Z(μτ)∩Z(ντ) ≠ ∅; say T ∈ ⋂_{τ∈S} Z(μτ) ∩ Z(ντ). For τ ≤ τ′ ∈ S, we have S′ ∩ Λ^{d(μτ)} = {μτ} for all S′ ∈ Z(μτ′). Hence T ∩ Λ^{d(μτ)} ⊆ {μτ} for all τ ∈ S. Since T ∩ Λ^{n} ≠ ∅ for all n, we deduce that T ∩ Λ^{d(μτ)} = {μτ} for all τ ∈ S. So ℓ_{μ}(S) ⊆ T, and since ℓ_{μ}(S) is an ultrafilter, the two are equal. Symmetry gives ℓ_{ν}(S) = T as well, so ℓ_{μ}(S) = ℓ_{ν}(S).

Proposition 7.7. Let Λ be a row-finite k-graph with no sources. Let c ∈ Z^{2}(ℤ^{k}, 𝕋) and suppose that (cc^{∗})|_{Per(Λ)} is nondegenerate. Suppose that t is a Cuntz-Krieger (Λ, c ∘ d)-family with each t_{v} nonzero. Suppose that a : H_{Per}Λ × H_{Per}Λ → ℂ is finitely supported. Then

∥ ∑_{μ,ν∈Λ} a_{μ,ν}t_{μ}t_{ν}^{∗}∥ ≥ ∥ ∑_{μ∼ν} a_{μ,ν}t_{μ}t_{ν}^{∗}∥.

Proof. Again as discussed at the beginning of the section, we may assume that c is a bicharacter. Let N = ⋁_{a_{μ,ν}≠0} d(μ). Using the Cuntz-Krieger relation, we can rewrite

∑_{μ,ν} a_{μ,ν}t_{μ}t_{ν}^{∗} = ∑_{μ,ν} ∑_{μμ′∈Λ^{N}} a_{μ,ν}c(μ, μ′)\overline{c(ν, μ′)}t_{μμ′}t_{νμ′}^{∗};

so we may assume without loss of generality that a_{μ,ν} ≠ 0 implies μ ∈ Λ^{N}. Since the t_{μ}t_{μ}^{∗} for μ ∈ Λ^{N} are mutually orthogonal, and since μ ∼ ν implies t_{μ}t_{μ}^{∗} = t_{ν}t_{ν}^{∗}, we have \overline{span}\{t_{μ}t_{ν}^{∗} : μ ∈ Λ^{N}, ν ∼ μ\} = ⨁_{μ∈Λ^{N}} \overline{span}\{t_{μ}t_{ν}^{∗} : μ ∼ ν\}, and so ∥ ∑_{μ∼ν} a_{μ,ν}t_{μ}t_{ν}^{∗}∥ = max_{μ∈Λ^{N}} ∥ ∑_{ν∼μ} a_{μ,ν}t_{μ}t_{ν}^{∗}∥. Choose μ_{max} such that

∥ ∑_{ν∼μ_{max}} a_{μ_{max},ν}t_{μ_{max}}t_{ν}^{∗}∥ ≥ ∥ ∑_{ν′∼μ′} a_{μ′,ν′}t_{μ′}t_{ν′}^{∗} ∥

for all μ′ ∈ Λ^{N}. Let v_{max} = s(μ_{max}). We claim that if μ, ν ∈ Λ satisfy s(μ) = s(ν) = v_{max} but μ ≁ ν, then there exists τ ∈ v_{max}Λ such that MCE(μτ, ντ) = ∅. Indeed, Lemma 7.6 implies that either there exists τ ∈ v_{max}Λ^{d(μ)∨d(ν)−d(μ)} such that μτ ∉ MCE(μ, ν) or there exists τ ∈ v_{max}Λ^{d(μ)∨d(ν)−d(ν)} such that ντ ∉ MCE(μ, ν). We consider the first case; the second is symmetric. We have μτ ∈ Λ^{d(μ)∨d(ν)}, and so μτ = αα′ for some α ∈ Λ^{d(ν)} \ {ν}. In particular, factoring μτ = ηζ with d(η) = d(ν), we have η ≠ ν, and so MCE(μτ, ντ) = ∅. An induction using that Λ has no sources now shows that there exists τ ∈ v_{max}Λ such that d(τ) > d(μ)∨d(ν) whenever a_{μ,ν} ≠ 0 and MCE(μτ, ντ) = ∅ whenever s(μ) = s(ν) = v_{max} and μ ≁ ν. Let l := d(τ). For μ, ν ∈ F with a_{μ,ν} ≠ 0, we have

∑_{λ∈F v_{max}∩Λ^{N}} t_{λτ} t_{λτ}^{∗} t_{μ}t_{ν}^{∗}t_{λτ} t_{λτ}^{∗}

= \begin{cases}c(N, l)\overline{c(d(ν), l)}t_{μτ} t_{ντ}^{∗} t_{μτ} t_{μτ}^{∗} & if s(μ) = v_{max} and μ ∼ ν \\ 0 & otherwise. \end{cases}

If μ ∼ ν then μτ ∼ ντ giving t_{μτ} t_{μτ}^{∗} = t_{ντ} t_{ντ}^{∗}, and hence t_{ντ}^{∗} t_{μτ} t_{μτ}^{∗} = t_{ντ}^{∗}. That is,

∑_{λ∈F v_{max}∩Λ^{N}} t_{λτ} t_{λτ}^{∗} t_{μ}t_{ν}^{∗}t_{λτ} t_{λτ}^{∗}

= \begin{cases}c(N, l)\overline{c(d(ν), l)}t_{μτ} t_{ντ}^{∗} & if s(μ) = v_{max} and μ ∼ ν \\ 0 & otherwise. \end{cases}

Since c is a bicharacter, the map m ↦ c(m, l) is a homomorphism of Per(Λ) into 𝕋. The universal property of A(c|_{Per(Λ)}) ensures that it admits an automorphism α_{l} such that α_{l}(U_{m}) = c(m, l)U_{m} for all m ∈ Per(Λ). An application of Proposition 7.5 with F = {μ_{max}τ} gives an injective homomorphism ρ_{1} : A(c|_{Per(Λ)}) → C^{∗}(t) which carries U_{N−d(ν)} to t_{μ_{max}τ} t_{ντ}^{∗} whenever ν ∼ μ_{max}; so ρ_{1} ∘ α_{l} carries U_{N−d(ν)} to ∑_{λ∈F v_{max}∩Λ^{N}} t_{λτ} t_{λτ}^{∗}. Proposition 7.5 applied to F = {μ_{max}} gives an injective homomorphism ρ_{2} : A(c|_{Per(Λ)}) → C^{∗}(t) which carries U_{N−d(ν)} to t_{μ_{max}}t_{ν}^{∗} whenever ν ∼ μ_{max}. Since these homomorphisms are injective, they are isometric, and so the map

ρ_{2} ∘ α_{l}^{−1} ∘ ρ_{1}^{−1} : \overline{span}\{t_{μ_{max}τ} t_{ντ}^{∗} : ν ∼ μ_{max}\} → \overline{span}\{t_{μ_{max}}t_{ν}^{∗} : ν ∼ μ_{max}\}

is isometric and carries spanning elements to spanning elements. Since the t_{λτ} t_{λτ}^{∗} are mutually orthogonal projections, the map a ↦ ∑_{λ} t_{λτ} t_{λτ}^{∗} at_{λτ} t_{λτ}^{∗} is norm-decreasing, and so we have

∥ ∑_{μ∼ν}a_{μ,ν}t_{μ}t_{ν}^{∗}∥

= ∥ ∑_{ν∼μ_{max}} a_{μ_{max},ν}t_{μ_{max}}t_{ν}^{∗}∥

= ∥ρ_{2} ∘ α_{l}^{−1} ∘ ρ_{1}^{−1}( ∑_{λ∈F v_{max}∩Λ^{n}} t_{λτ} t_{λτ}^{∗} ( ∑_{ν∼μ_{max}} a_{μ_{max},ν}t_{μ_{max}}t_{ν}^{∗})t_{λτ} t_{λτ}^{∗} )∥

= ∥ ∑_{λ∈F v_{max}∩Λ^{n}} t_{λτ} t_{λτ}^{∗} ∑_{μ,ν∈Λ} a_{μ,ν}t_{μ}t_{ν}^{∗}t_{λτ} t_{λτ}^{∗} ∥

≤ ∥ ∑_{μ,ν∈Λ} a_{μ,ν}t_{μ}t_{ν}^{∗}∥,

as required.

Proof of Theorem 7.1. Since Λ is cofinal, the saturation of H_{Per} is all of Λ^{0}. Hence Lemma 4.3 implies that P_{H_{Per}}C^{∗}(Λ, c∘d)P_{H_{Per}} is a full corner of C^{∗}(Λ, c∘ d). So it suffices to show that P_{H_{Per}}C^{∗}(Λ, c ∘ d)P_{H_{Per}} is simple. Let Γ = H_{Per}Λ. The gauge-invariant uniqueness theorem gives a canonical isomorphism C^{∗}(Γ, c ∘ d) ≅ P_{H_{Per}}C^{∗}(Λ, c ∘ d)P_{H_{Per}}. So it suffices to show that C^{∗}(Γ, c ∘ d) is simple. Suppose π is a homomorphism C^{∗}(Γ, c ∘ d) → B, and let s := {s_{λ} : λ ∈ Γ} be the universal Cuntz-Krieger (Γ, c ∘ d)-family. First suppose that π(s_{v}) = 0 for some v. Then ker(π) contains the gauge-invariant ideal generated by s_{v}, which in turn, by Theorem 4.6, contains s_{w} for every w in the saturated hereditary set generated by v. Since Γ is cofinal it follows that π(s_{w}) = 0 for all w, and then π(s_{μ}s_{ν}^{∗}) = π(s_{r(μ)}s_{μ}s_{ν}^{∗}) = 0 for all μ, ν, and hence π = 0. Now suppose that π(s_{v}) ≠ 0 for all v; we must show that π is injective. Let t_{μ} := π(s_{μ}) for all μ. Since H_{Per} is all of Γ^{0}, Proposition 7.7 implies that ∥ ∑ a_{μ,ν}t_{μ}t_{ν}^{∗}∥ ≥ ∥ ∑_{μ∼ν} a_{μ,ν}t_{μ}t_{ν}^{∗}∥ for all finitely supported a, and so there is a norm-decreasing linear map Ψ : C^{∗}(t) → \overline{span}\{t_{μ}t_{ν}^{∗} : μ ∼ ν\} such that Ψ(t_{μ}t_{ν}^{∗}) = t_{μ}t_{ν}^{∗} if μ ∼ ν and is zero otherwise. The same argument gives a norm-decreasing linear map Φ from C^{∗}(Γ, c ∘ d) to \overline{span}\{s_{μ}s_{ν}^{∗} : μ ∼ ν\}. This is faithful on positive elements because the faithful conditional expectation Θ of Proposition 7.3 satisfies Θ = Θ ∘ Ψ. Clearly π ∘ Φ = Ψ ∘ π. Fix a finite linear combination ∑_{μ∼ν} a_{μ,ν}s_{μ}s_{ν}^{∗}. Using the Cuntz-Krieger relation as in the proof of Proposition 7.7, we may assume that there exists n ∈ ℕ^{k} such that a_{μ,ν} ≠ 0 implies that μ ∈ Λ^{n}. Applying Proposition 7.5 to the Cuntz-Krieger (Γ, c ∘ d)-families s and t and with F = {μ : a_{μ,ν} ≠ 0} and composing the resulting homomorphisms, we obtain an isometric map from C^{∗}({s_{μ}s_{ν}^{∗} : λ ∈ F, μ ∼ λ ∼ ν}) to C^{∗}({t_{μ}t_{ν}^{∗} : λ ∈ F, μ ∼ λ ∼ μ}) which carries each s_{μ}s_{ν}^{∗} to t_{μ}t_{ν}^{∗}. Hence ∥ ∑_{μ∼ν} a_{μ,ν}s_{μ}s_{ν}^{∗}∥ = ∥ ∑_{μ∼ν} a_{μ,ν}t_{μ}t_{ν}^{∗}∥. Since a was arbitrary, we deduce that π|_{\overline{span}\{s_{μ}s_{ν}^{∗}:μ∼ν\}} is injective. So Lemma 3.14 shows that π is injective.

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Aidan Sims, Benjamin Whitehead, Michael F. Whittaker School of Mathematics and Applied Statistics The University of Wollongong NSW 2522 AUSTRALIA