GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 4, 1998, 333-340

ON THE UNIVERSAL C^{∗}-ALGEBRA GENERATED BY PARTIAL ISOMETRY

T. KANDELAKI

Abstract. A universal C^{∗}-algebra is constructed which is generated by a partial isometry. Using grading on this algebra we construct an analog of Cuntz algebras which gives a homotopical interpretation of KK-groups. It is proved that this algebra is homotopy equivalent up to stabilization by 2×2 matrices to M_{2}(C). Therefore those algebras are KK-isomorphic.

We recall the definition of partial isometry.

Definition 1. Let H_{1} and H_{2} be Hilbert spaces and v : H_{1} ⟶ H_{2} be a bounded linear map. v is called a partial isometry if v^{∗}v is a projection.

We only want to emphasize the following important facts. The projection p = v^{∗}v is called the support projection for v. Standard equations involving partial isometries are the following:

v = vv^{∗}v = vp = qv, v^{∗} = v^{∗}vv^{∗} = pv^{∗} = v^{∗}q.

It is known that the above equations can be taken to define a partial isometry.

First of all, our aim is to construct an involutive algebra generated by one symbol together with the above relations and prove the existence of a maximal C^{∗}-norm on it. For the general construction and examples see [1].

Let U(v) be the universal involutive complex algebra generated by the symbols v, v^{∗}. Let J(v) be the two-sided ∗-ideal generated by elements of the following form:

(a) {v^{∗}} − {v}^{∗}, (b) {v} − {vv^{∗}v}, where {v} and {v^{∗}} denote the elements of U(v) which correspond to v and v^{∗} respectively. Then U(v) will denote the factor algebra U(v)/J(v), which is a complex ∗-algebra and has the universal property that if B is a

333 ∗-algebra generated by a partial isometry ν, then there exists a canonical ∗-homomorphism κ : U(v) → B such that κ(v) = ν.

1991 Mathematics Subject Classification. 19K, 46L.

Key words and phrases. The universal C^{∗}-algebra of a partial isometry, KK-groups, homotopy up to stabilization.

1072-947X/98/0700-0333$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

We say that an algebra seminorm p on a complex involutive algebra A is a C^{∗}-seminorm if p(xx^{∗}) = p(x)^{2}. If ∥x∥ = sup{p(x) | p is a C^{∗}- seminorm on A} is finite for every x ∈ A, then ∥·∥ defines the largest possible C^{∗}-seminorm on A, and we can form the completion C^{∗}(A) = \overline{A/N^{∥·∥}}, N = {x ∈ A | ∥x∥ = 0}. C^{∗}(A) has the universal property that any ∗- homomorphism from A into a C^{∗}-algebra factors uniquely through the map A → C^{∗}(A) (which is not necessarily injective) [1].

Proposition 2. There exists a C^{∗}-seminorm ∥ · ∥ on U(v) (which is the supremum of all C^{∗}-seminorms on the same algebra). Let C^{∗}(v) = C^{∗}(U(v)) and κ : U(v) → C^{∗}(v) be the canonical ∗-homomorphism; then C^{∗}(v) has the following universal property: if φ : U(v)→ B is a ∗-homo-morphism into a C^{∗}-algebra B, then there exists a unique ∗-homomorphism ψ : C^{∗}(v) → B such that the diagram

U(v) →^{κ} C^{∗}(v)
↘^{φ} ↓^{ψ}
B

is commutative.

Proof. Let θ be any C^{∗}-seminorm on U(v); then θ(v)^{2} = θ(v^{∗}v) ≤ 1 because v^{∗}v is a projection. If e ∈ U(v), then e = ∑_{i} λ_{i}f_{i_{1}} ⋯ f_{i_{n_{i}}}, where f_{i_{k}}

is v or v^{∗}. Thus θ(e) ≤ ∑_{i} |λ_{i}|θ(f_{i_{1}}) ⋯ θ(f_{i_{n_{1}}} ) ≤ ∑_{i} |λ_{i}|. So there exists

π(e) = sup_{θ} ∥θ(e)∥, where θ runs over all C^{∗}-seminorms on U(v). It is easy to check that π is a C^{∗}-seminorm and C^{∗}(v) = C^{∗}(U(v)) has the above universal property.

The algebra C^{∗}(v) has a Z_{2}-grading which is induced by the automorphism defined by v → −v; this graded algebra will be denoted by G^{∗}(v). It follows from the definition that deg v = 1. If B is a Z_{2}-graded C^{∗}-algebra and υ is a partial isometry with deg υ = 1, then there exists a unique graded ∗-homomorphism ψ : G^{∗}(v) → B such that ψ(v) = υ.

Now we need a definition of KK-groups in the style given in [2].

Definition 3. A superquasimorphism (sqm) from A to B is a triple Φ = (ϕ, G, μ), where ϕ is a graded homomorphism from A to a Z_{2}-graded algebra D with a graded (invariant) ideal J ◁ D and G ∈ D is an element of degree 1, such that

(G − G^{∗})ϕ(x) ∈ J, (1 − G^{2})ϕ(x) ∈ J, [ϕ(x), G] ∈ J

ON THE UNIVERSAL C^{∗}-ALGEBRA 335

for x ∈ A, μ : A → B is a homomorphism. This will be writtten as

A →^{ϕ} D ▷^{G} J →^{μ} B or shortly Φ : A − − ▷ B.

A mapping between two sqm’s Φ_{1}, Φ_{2} is a commutative diagram

\begin{matrix}A & →^{ϕ_{1}} & D_{1} & ▷^{G_{1}} & J_{1} & →^{μ_{1}} & B \\ || & & ↓ & & ↓ & & || \\ A & →^{ϕ_{2}} & D_{2} & ▷^{G_{2}} & J_{2} & →^{μ_{2}} & B \end{matrix} ;

D_{1} → D_{2} maps G_{1} to G_{2}. We will say that Φ_{1} is contained in Φ_{2} if the vertical homomorphisms are injective and that Φ_{2} is a quotient of Φ_{1} if they are surjective.

If f : A′ → A, g : B → B′ are homomorphisms and Φ : A − − ▷ B is a sqm then the composition g ∘ Φ ∘ f gives a sqm from A′ to B′.

Let q_{t} : B[0; 1] → B be the evolution at time t. A sqm Π from A to B[0; 1] will be called a homotopy from Φ to Ψ, if Φ = q_{0} ∘ Π, Ψ = q_{1} ∘ Π. Φ and Ψ will be called equivalent if there is a chain {Φ_{i,}i = 0, …, n} consisting of finitely many sqm Φ_{i,} such that Φ_{0} = Φ, Φ_{n} = Ψ and, for each i, Φ_{i} and Φ_{i+1} are connected either by a mapping or by a homotopy [2].

Following [2], let us denote by KK(A, B) the set of equivalence classes of

sqm’s from A to K^{∧} ⊗B, where K^{∧}=M_{2}^{∧} (K) with the standard even grading

on M^{∧}_{2}, and K is the algebra of compact operators on a separable Hilbert

space. Choose a fixed (standard) isomorphism K^{∧} ⊗ K^{∧}=K^{∧} such that the

identity id_{K^{∧}} is homotopic to the embedding j_{0} mapping x ∈K^{∧} to e ⊗ x with e a minimal projection of degree 0 in the left upper corner.

On this set there is a (commutative) addition defined by

[Φ_{1}] ⊕ [Φ_{2}] = [Φ_{1} ⊕ Φ_{2}], Φ_{1} ⊕ Φ_{2} : A − − ▷ M_{2}(K^{∧} ⊗B) ≃K^{∧} ⊗B.

The result will not depend up to an equivalence on the chosen isomorphism

M_{2}(K^{∧} ⊗B) ≃K^{∧} ⊗B.

It follows from the definition that every sqm is contained in a sqm with

unital D. Let Φ =A →^{ϕ} D ▷^{G} J →^{μ} B be a sqm, where A and D are separable and D is unital. Let

D′ = {d ∈ D | [ϕ(x), d] ∈ J,for all x ∈ A} and

J′ = {d ∈ D | d · ϕ(x) ∈ J, ϕ(x) · d ∈ J, for all x ∈ A}. It is easy to check that D′ is a unital C^{∗}-algebra and J′ is a closed two-sided ideal in D′. Note that G ∈ D′, G − G^{∗} ∈ J′, and 1 − G^{2} ∈ J′. Thus G gives G′, the unitary element in the factor algebra D′/J′. It is known that every element of a factor C^{∗}-algebra can be lifted to the C^{∗}-algebra preserving the norm [3], [4]. Thus G′ can be lifted to D′ preserving the norm, i.e., there exists an element F such that F − G ∈ J′ and ||F || = 1. The matrix

U_{F} = ( \begin{matrix}F & −(1 − F F^{∗})^{1/2} \\ (1 − F^{∗}F)^{1/2} & F^{∗} \end{matrix} )

is a unitary element in the M_{2}(D′) and M_{2}(D). Thus we have the following sqm’s

Φ_{k} = A →^{i_{0}ϕ} M_{2}(D) ▷^{U_{k}} M_{2}(J) →^{μ} M_{2}(B), k = 0, 1,

where U_{0} = (\begin{matrix}F & 0 \\ 0 & 0 \end{matrix}), U_{1} = U_{F}, i_{0} is an inclusion in the upper left corner.

It is easy to see that Φ_{0} ∼ Φ_{1} because (a) (1 − F^{∗}F)^{1/2}ϕ(x) and (1 − F F^{∗})^{1/2}ϕ(x) are in J, for all x ∈ A;

(b) U_{F} is homotopic to U_{d} = (\begin{matrix}F & 0 \\ 0 & F^{∗} \end{matrix}) .

Therefore we have

Lemma 4. Every sqm A →^{ϕ} D ▷^{G} J →^{μ} B, with unital D, is equivalent

up to 2 × 2 matrices to sqm A →^{i_{0}ϕ} M_{2}(D) ▷^{U} M_{2}(J) →^{μ} M_{2}(B), where U is a unitary element in M_{2}(D).

Corollary 5. Every sqm is equivalent up to 2×2 matrices to a sqm with G a partial isometry.

This corollary can be made more precise if A and D are separable. It is known that if A → B is an epimorphism of separable C^{∗}-algebras then any unitary element of B can be lifted to a partial isometry of A [4]. Thus G′ can be lifted to a parial isometry Υ ∈ D′ because D′ is separable. Note that (G − Υ)ϕ(x) ∈ J, [ϕ(x), Υ] ∈ J. Elementary calculation shows that

Φ′ = A →^{ϕ} D ▷^{Υ} J →^{μ} B is a superquasimorphism, which is equivalent to Φ

by the homotopy Π = A →^{ϕ_{c}} D[0; 1] ▷^{H} J[0; 1] →^{ρ} B[0; 1], where ϕ_{c}(x)(t) = ϕ(x), H(t) = G − t(G − Υ) and ρ(f)(t) = μ(f(t)). Thus we get

Lemma 6. Every sqm A →^{ϕ} D ▷^{G} J →^{μ} B with separable A is equivalent

to the sqm A → D_{1} ▷^{Υ} J_{1} → B, where Υ is a partial isometry.

Therefore we have two variants of the definition of KK-groups:

(1) Consider the definition of sqm’s with partial isometry as G. Then we get the same KK-groups. In this case we have the following universal sqm:

Let A ∗ C^{∗}(v) be the sum of C^{∗}-algebras in the category of C^{∗}-algebras and ∗-homomorphisms (it is exactly the free product of C^{∗}-algebras). Let v(A) be the closed two-sided ideal in A ∗ C^{∗}(v) generated by the elements (v − v^{∗}) · a, (1 − v^{2}) · a, [a, v]. Then we have the sqm A →^{i_{0}} A ∗ C^{∗}(v) ▷^{v} v(A) which is universal in the following sense: if A →^{ϕ} D ▷^{G} J → B^{μ} is a given sqm, where G is a partial isometry, then there exist a unique ∗-homomorphism

v(A) →^{v(μ)} B and a mapping

A →^{i_{0}} A ∗ C^{∗}(v) ▷^{v} v(A) →^{v(μ)} B
||  ↓ ϕ_{g}  ↓ ϕ_{g}  ||
A →^{ϕ}  D  ▷^{G} J  →^{μ} B

where ϕ_{g} | A = ϕ and ϕ_{g} | C^{∗}(A)(v) = G. Therefore we have

Theorem 7. KK(A, B) = [v(A), K^{∧} ⊗B], where [v(A), K^{∧} ⊗B] denotes the set of homotopy classes of (graded) homomorphisms.

Theorems of such a type come from [5]; see also [2].

(2) Consider a sqm with unital algebras D and unitary element G ∈ D. In this case we have the following universal sqm:

Let C^{∧} (S^{1}) be the standard C^{∗}-algebra of the circle with grading induced by the automorphism z → −z and let A^{+} be obtained from A by adjoining

the unit. Let A^{+}• C^{∧} (S^{1}) be the sum in the category of unital C^{∗}-algebras and unital homomorphisms (this is the factor algebra of the free product by the ideal generated by the element 1_{A} − 1_{S^{1}}) and define z(A) as the closed ideal generated by the elements [a, z], (z − z^{∗})a, a ∈ A; then we have

the universal sqm A →^{i_{A}} A^{+}• C^{∧} (S^{1}) ▷^{z} z(A) with the following property: if Φ is a sqm with unital D and unitary element G, then there exists a unique homomorphism z(μ) : z(A) → B such that the diagram

A →^{i_{A}} A^{+}• C^{∗}^{∧} (S^{1}) ▷^{z} z(A) →^{z(μ)} B
||  ↓ ϕ_{z}  ↓  ||
A →^{ϕ}  D  ▷^{G} J  →^{μ} B

is commutative, where ϕ_{z} = ϕ^{+} • g_{z} and g_{z} :C^{∗}^{∧} (S^{1}) → D is the homomorphism defined by the formula g_{z}(z) = G (because G is a unitary element in D). We have the following analog of Theorem 7:

Theorem 8. KK(A, B) = [z(A), K^{∧} ⊗B].

We give below some homotopic and KK-theoretic properties of C^{∗}- algebras C^{∗}(v) and G^{∗}(v).

Remark. Let S be the operator on the Hilbert space l_{2}(N) given on the basis by e_{n} → e_{n_{+1}} (it is called the unilateral shift). The Toeplitz algebra T is the unital separable C^{∗}-subalgebra of l_{2}(N) generated by S. The algebra T is the universal C^{∗}-algebra generated by an isometry [1]. The algebra T has the grading defined by the map S ⟼ −S. We denote this graded Toeplitz algebra by T̂. The C^{∗}-algebras C^{∗}(v) and G^{∗}(v) are sufficiently “large” algebras, because the universal property implies that the canonical ∗-homomorphisms ν : C^{∗} → T and υ : G^{∗}(A) → T̂ are ∗-epimorphisms.

We will need the folowing ∗-homomorphisms: i_{0} : M_{2}(C) → M_{2}(M_{2}(C)) and j_{0} : C^{∗}(v) → M_{2}(C^{∗}(v)) which are ∗-inclusions into the upper left corner.

We recall that two ∗-homomorphisms f_{0}, f_{1} : A → B of C^{∗}-algebras are homotopic if there is a path {φ_{t}} of ∗-homomorphisms φ_{t} : A → B such that t ⟼ φ_{t}(a) is a norm continuous map from [0; 1] to B for fixed a ∈ A and such that φ_{0} = f_{0} , φ_{1} = f_{1}.

Now we are ready to prove

Theorem 9. Let C^{∗}(v) be the universal C^{∗}-algebra of a partial isometry. There exists a ∗-homomorphism φ_{v} : M_{2}(C) → M_{2}(C^{∗}(v)) such that φ_{v}μ is homotopic to j_{0}, and such that μφ_{v} is homotopic to i_{0} (here μ stands for both the evolution map M_{2}(C^{∗}(v)) → M_{2}(M_{2}(C)) and the one C^{∗}(v) → M_{2}(C)). That is, C^{∗}(v) and M_{2}(C) are homotopy equivalent up to stabilization by 2 × 2 matrices.

Proof. Let φ_{v} : M_{2}(C) → M_{2}(C^{∗}(v)) be defined by the formula

(\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) ⟼ (\begin{matrix}0 & v \\ 0 & 0 \end{matrix}) .

There is homotopy h(t) = (\begin{matrix}cos t · v & sin t · v \\ 0 & 0 \end{matrix}) from (\begin{matrix}v & 0 \\ 0 & 0 \end{matrix}) to (\begin{matrix}0 & v \\ 0 & 0 \end{matrix}),

where t ∈ [0; 1]. Note that h(t) is a partial isometry in M_{2}(C^{∗}(v), for any t ∈ [0; 1] :

h(t) · h(t)^{∗} (\begin{matrix}cos t · v & sin t · v \\ 0 & 0 \end{matrix}) (\begin{matrix}cos t · v^{∗} & 0 \\ sin t · v^{∗} & 0 \end{matrix}) = (\begin{matrix}vv^{∗} & 0 \\ 0 & 0 \end{matrix}) .

Thus we have, by the universal property of C^{∗}(v), a path of ∗-homomor-phisms: H_{t} : C^{∗}(v) → M_{2}(C^{∗}(v)) which is defined by the formula

H_{t}(v) = (\begin{matrix}cos t · v & sin t · v \\ 0 & 0 \end{matrix})

such that H_{0} = j_{0} and H_{1} = φ_{0}μ. We have to prove now that μφ_{0} is homotopic to i_{0}. Note that μφ_{0} is defined by the formula

(\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) ⟼ (\begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) .

The elements

v′ = (\begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) and v′′ = (\begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix})

are unitary equivalent with the unitary element

u = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & −1 & 0 & 0 \end{matrix}) ,

i.e., u^{∗}v′u = v′′. The element u is homotopic to 1 in the group of unitary elements, by the homotopy

u_{t} = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & cos \frac{π}{2} t & 0 & sin \frac{π}{2} t \\ 0 & 0 & 1 & 0 \\ 0 & − sin \frac{π}{2} t & 0 & cos \frac{π}{2} t \end{matrix}) ,

t ∈ [0; 1]. Thus the path of ∗-homomorphisms H′_{t} : M_{2}(C) → M_{2}(M_{2}(C) defined by the formula H′_{t}(x) = u_{t}^{∗}i_{0}u_{t} defines the homotopy from i_{0} to μφ_{v}.

As an application of the theorem we now have to prove

Corollary 10. The canonical ∗-homomorphism μ : C^{∗}(v) → M_{2}(C) induces an invertible element in KK(C^{∗}(v); M_{2}(C)).

Proof. It follows from the definition of KK-groups that i_{0} and j_{0} induce identity elements of rings KK(C^{∗}(v), C^{∗}(v)) and KK(M_{2}(C), M_{2}(C)). Homomorphisms μ and φ_{v} give elements of the groups KK(C^{∗}(v), M_{2}(C)) and KK(M_{2}(C); C^{∗}(v)), respectively. From the theorem it immediately follows that [φ_{v}] · [μ] = [1_{C^{∗}(v)}] and [μ] · [φ_{v}] = [1_{M_{2}(C)}].

Thus C^{∗}-algebras C^{∗}(v) and M_{2}(C) are KK-isomorphic.

Let A be a Z_{2}-graded C^{∗}-algebra. On M_{2}(A) we have two essential Z_{2}-gradings given by ∗-automorphisms of period 2:

(\begin{matrix}a & b \\ c & d \end{matrix}) ⟼^{γ_{0}} (\begin{matrix}γ(a) & γ(b) \\ γ(c) & γ(d) \end{matrix}) and (\begin{matrix}a & b \\ c & d \end{matrix}) ⟼^{γ_{1}} (\begin{matrix}γ(a) & −γ(b) \\ −γ(c) & γ(d) \end{matrix})

where γ : A → A is the ∗-automorphism inducing the given Z_{2}-grading on A. Let M_{2}(A) be the algebra of 2 × 2 matrices with an odd grading and

M^{∧}_{2} (C) be the same algebra with an even grading. The partial isometry

e_{12} = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) has degree 1 in the odd grading and so we have the canonical Z_{2}-graded ∗-homomorphism ω : G^{∗}(v) →M_{2}^{∧} (C) defined by the formula ω(v) = e_{12,} which follows from the universal property of G^{∗}(v). It follows from the above definitions of Z_{2}-gradings that the canonical ∗-inclutions

i_{0} :M_{2}^{∧} (C) → M_{2}(M_{2}(^{∧} C)) and j_{0} : G^{∗}(v) → M_{2}(G^{∗}(v))

are graded homomorphisms.

We have the following graded analog of Theorem 9.

Theorem 11. Let G^{∗}(v) be the universal Z_{2}-graded algebra of a partial

isometry of degree one. There exists a graded ∗-homomorphism φ_{v} :M_{2}^{∧} (C) → M_{2}(G^{∗}(v)) such that ωφ_{v} is homotopic to i_{0} and φ_{v}ω is homotopic

to j_{0}. That is, G^{∗}(v) and M_{2}(C)^{∧} are homotopy equivalent up to stabilization by odd graded 2 × 2 matrices.

Proof. It exactly coincides with the proof of Theorem 9.

Corollary 12. The canonical ∗-homomorphism ω induces an invertible

element of KK(G^{∗}(v), M_{2}^{∧} (C)).

Acknowledgement

This research was supported in part by ISF grant No. MXH200 and INTAS grant No. 93-2618. The author would like to thank J. Cuntz and U. Haag for their papers.

References

1. J. Cuntz, A survey of some aspects of non-commutative geometry. Jber. d. Dt. Math.-Verein 95(1993), 60–84.

2. U. Haag, Cuntz picture in ℤ/2ℤ-graded KK-theory. Ph.D. thesis at the University of Heidelberg, 1994.

3. C. A. Akemann and G. K. Pedersen, Ideal perturbations of elements in C^{∗}-algebras. Math. Scand. 41(1977), 117–138.

4. N. E. Wegge-Olsen, K-theory and C^{∗}-algebras. Oxford University Press, Oxford, New York, Tokyo, 1993.

5. J. Cuntz, A new look at KK-theory. K theory 1(1987), 31–51.

(Received 15.12.1995; revised 15.01.1997)

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093

Georgia