Simple tracially 𝒵-absorbing C*-algebras

Journal of Geometry and Physics, 2020

Following Elliott's earlier work, we show that the Elliott invariant of any finite separable simple C *-algebra with finite nuclear dimension can always be described as a scaled simple ordered group pairing together with a countable abelian group (which unifies the unital and nonunital, as well as stably projectionless cases). We also show that, for any given such invariant set, there is a finite separable simple C *-algebra whose Elliott invariant is the given set, a refinement of the range theorem of Elliott in the stable case. In the stably projectionless case, modified model C *-algebras are constructed in such a way that they are of generalized tracial rank one and have other technical features. We also show that every stably projectionless separable simple amenable C *-algebra in the UCT class has rationally generalized tracial rank one.

Operator Algebras and Their Applications

We prove that every unital stably finite simple amenable C *-algebra A with finite nuclear dimension and with UCT such that every trace is quasi-diagonal has the property that A ⊗ Q has generalized tracial rank at most one, where Q is the universal UHF-algebra. Consequently, A is classifiable in the sense of Elliott. In particular, if A is a unital separable simple C *-algebra with finite decomposition rank which satisfies the UCT, then A is classifiable. Definition 2.2. (N. Brown [5]) Let A be a unital C *-algebra. Denote by T (A) the tracial state space of A, and denote by T qd (A) the tracial states τ with the following property: For any finite subset F and ε > 0, there exists a contractive completely positive linear map ϕ : A → Q such that |τ (a) − τ (ϕ(a))| < ε for all a ∈ F and (2.1) ϕ(a)ϕ(b) − ϕ(ab) < ε for all a, b ∈ F. (2.2) Definition 2.3. Let F 1 and F 2 be two finite dimensional C *-algebras and let ψ 0 , ψ 1 : F 1 → F 2 be two unital homomorphisms. Denote by C = C(F 1 , F 2 , ψ 0 , ψ 1) = {f ∈ C([0, 1], F 2) ⊕ F 1 : f (0) = ψ 0 (f) and f (1) = ψ 1 (f)}. Let C be the class of unital C *-algebras of the above from. C *-algebras in the class are called Elliott-Thomsen building blocks. These are also called 1-dimenional non-commutative CW complex (NCCW). Denote by C 0 the subclass of C consisting of those C *-algebras C ∈ C such that K 1 (C) = {0}. Definition 2.4. (see 9.1 of [8]) Let A be a unital infinite dimensional simple C *-algebra. We say A has generalized tracial rank at most one, if the following hold: Let ε > 0, let a ∈ A + \ {0} and let F ⊂ A be a finite subset. There exists a nonzero projection p ∈ A and a C *-subalgebra C ∈ C with 1 C = p such that xp − px < ε for all x ∈ F, (2.3) dist(pxp, C) < ε for all x ∈ F and (2.4)

Mathematische Annalen, 2013

We give an example of an exact, stably finite, simple, separable C*-algebra D which is not isomorphic to its opposite algebra. Moreover, D has the following additional properties. It is stably finite, approximately divisible, has real rank zero and stable rank one, has a unique tracial state, and the order on projections over D is determined by traces. It also absorbs the Jiang-Su algebra Z, and in fact absorbs the 3 ∞ UHF algebra. We can also explicitly compute the K-theory of D, namely K0(D) ∼ = Z 1 3 with the standard order, and K1(D) = 0, as well as the Cuntz semigroup of D, namely W (D) ∼ = Z 1 3 + ⊔ (0, ∞).

Illinois Journal of Mathematics, 2014

In this paper we define a Rokhlin property for automorphisms of nonunital C *-algebras and for endomorphisms. We show that the crossed product of a C *-algebra by a Rokhlin automorphism preserves absorption of a strongly self-absorbing C *-algebra, and use this result to deduce that the same result holds for crossed products by endomorphisms in the sense of Stacey. This generalizes earlier results of the second named author and W. Winter.

We prove that every unital stably finite simple amenable $C^*$-algebra $A$ with finite nuclear dimension and with UCT such that every trace is quasi-diagonal has the property that $A\otimes Q$ has generalized tracial rank at most one, where $Q$ is the universal UHF-algebra. Consequently, $A$ is classifiable in the sense of Elliott. In particular, if $A$ is a unital separable simple $C^*$-algebra with finite decomposition rank which satisfies the UCT, then $A$ is classifiable.

arXiv (Cornell University), 2019

A classification theorem is obtained for a class of unital simple separable amenable Zstable C *-algebras which exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable Z-stable C *-algebras. Moreover, it contains all unital simple separable amenable C *-algebras which satisfy the UCT and have finite rational tracial rank. Contents 22 Construction of maps 2 23 A pair of almost commuting unitaries 21 24 More existence theorems for Bott elements 26 25 Another Basic Homotopy Lemma 32 26 Stable results 40 27 Asymptotic unitary equivalence 46 28 Rotation maps and strong asymptotic equivalence 60 Note that B is a unital simple AH-algebra with no dimension growth, with real rank zero, and with a unique tracial state. Also h := ψ 1,∞ : C → B gives [h]| K(C) (z) = z ⊗ 1 D for z ∈ K(C) and [h]| K(C 0) (x) = x ⊗ [1 U 1 ] for x ∈ K(C 0). Let η 0 : C 0 ⊗ U 1 → C 0 ⊗ U be defined by η 0 := id C 0 ⊗ η. Define κ ∈ KL(B, B) by κ| K(C⊗U) = id K(C⊗U) and κ| K(C 0 ⊗U 1) = η 0. Note that κ ∈ KL(B, B) ++. Recall K(C ⊗ U) = K(C ⊗ U) ⊕ K(C 0 ⊗ U). One may also view κ as an element in Hom Λ (K(C ⊗ U) ⊕ K(C 0 ⊗ U 1), K(C ⊗ U)) = KL(B, C ⊗ U). It has an inverse κ −1 ∈ KL(C ⊗ U, B). We have κ −1 • [j C ] = [h]. Note that 1 − ψ n,∞ (e n) commutes with the image of h for all n ≥ N. Moreover, (1 − ψ n,∞ (e n))h(c)(1 − ψ n,∞ (e n)) = ψ n,∞ ((1 − e n)ψ N,n • ϕ 1,N (c)(1 − e n)) for all c ∈ C. Therefore the map (1 − ψ n,∞ (e n))h(C)(1 − ψ n,∞ (e n)) has finite dimensional range. We also haveα 1 • κ ∈ KL e (B, A) ++ , where (recall) A = A 1 ⊗ U. We also note that B has a unique tracial state. Let γ : T (A) → T (B) be defined by γ(τ) = t 0 where t 0 ∈ T (B) is the unique tracial state. It follows thatα 1 • κ and γ are compatible. By Corollary 21.11 of [21], there is a unital homomorphism H : B → A such that [H] =α 1 • κ. Define ϕ : C → A by ϕ = H • h. Then, ϕ is injective, and, by (e 22.5) and [h] = κ −1 • [j C ] , we have [ϕ] = α. To show the last part, define q n = ψ n+1,∞ (e n) ∈ B, n = N + 1, N + 2, .... Define p n = 1 − H(q n), n = N + 1, N + 2, .... One checks that lim n→∞ max{τ (1 − p n) : τ ∈ T (A)} = lim n→∞ l n m n = 0. (e 22.6) Note that for n > N , q n commutes with the image of h and the homomorphism (1−q n)h(1−q n) : C → (1 − q n)B(1 − q n) has finite dimensional range. Define ϕ ′ n : C → (1 − p n)A(1 − p n) by ϕ ′ n (f) = H(q n)H • h(f)H(q n) for f ∈ C. Define ϕ ′′ n (f) = (1 − p n)H • h(f)(1 − p n), which is a point-evaluation map. The lemma follows. We also have the following: Lemma 22.2. Let C = M k (C(T)) and let A be a unital infinite dimensional simple C *-algebra with stable rank one and with the property (SP). Then the conclusion of 22.1 also holds for a given α ∈ KK e (C, A) ++. Proof. Let p 1 ∈ C be a minimal rank one projection. Since kα([p 1 ]) = α([1 C ]) = [1 A ], A contains mutually equivalent and mutually orthogonal projections e 1 , e 2 , ..., e k such that k i=1 e i = 1 A. Thus A = M k (A ′), where A ′ ∼ = e 1 Ae 1. Since e i Ae i are unital infinite dimensional simple C *algebras with stable rank one and with (SP), the general case can be reduced to the case that k = 1. Fix 1 > δ > 0. Choose a non-zero projection p ∈ A such that τ (p) < δ for all τ ∈ T (A). Note K 1 (pAp) = K 1 (A), since A is simple. Let α 1 : K 1 (C(T)) → K 1 (pAp) be the homomorphism given by α. Let z ∈ C(T) be the standard unitary generator. Let x = α 1 ([z]) ∈ K 1 (pAp). Since pAp has stable rank one, there is a unitary u ∈ pAp such that [u] = x in K 1 (pAp) = K 1 (A). Define ϕ ′ : C(T) → pAp by ϕ ′ (f) = f (u) for all f ∈ C(T). Define ϕ ′′ : C(T) → (1 − p)A(1 − p) by ϕ ′′ (f) = f (1)(1 − p) for all f ∈ C(T) (where f (1) is the point evaluation at 1 on the unit circle). Define ϕ = ϕ ′ ⊕ ϕ ′′ : C(T) → A. The map ϕ verifies the conclusion of lemma follows. Corollary 22.3. Let X be a connected finite CW complex, let C = P M m (C(X))P, where P ∈ M m (C(X)) is a projection, let A 1 ∈ B 0 be a unital separable simple C *-algebra which satisfies the UCT, and let A = A 1 ⊗ U, where U is a UHF-algebra of infinite type. Suppose that α ∈ KK e (C, A) ++ and γ : T (A) → T f (C(X)) is a continuous affine map. Then there exists a sequence of contractive completely positive linear maps h n : C → A such that

Documenta Mathematica, 2002

tn i=1 M [n,i] (C(X n,i)), φ n,m) is a simple C *-algebra, where X n,i are compact metrizable spaces of uniformly bounded dimensions (this restriction can be relaxed to a condition of very slow dimension growth). It is proved in this article that A can be written as an inductive limit of direct sums of matrix algebras over certain special 3-dimensional spaces. As a consequence it is shown that this class of inductive limit C *-algebras is classified by the Elliott invariant-consisting of the ordered K-group and the tracial state space-in a subsequent paper joint with G. Elliott and L. Li (Part II of this series). (Note that the C *-algebras in this class do not enjoy the real rank zero property.) Guihua Gong Contents §0 Introduction. §1 Preparation and some preliminary ideas. §2 Spectral multiplicity. §3 Combinatorial results. §4 Decomposition theorems. §5 Almost multiplicative maps. §6 The proof of main theorem.

2018

We prove simplicity of all intermediate $C^*$-algebras $C^*_{r}(\\Gamma)\\subseteq \\mathcal{B} \\subseteq \\Gamma\\ltimes_r C(X)$ in the case of minimal actions of $C^*$-simple groups $\\Gamma$ on compact spaces $X$. For this, we use the notion of stationary states, recently introduced by Hartman and Kalantar (arXiv:1712.10133v2). We show that the Powers&#39; averaging property holds for the reduced crossed product $\\Gamma\\ltimes_r \\mathcal{A}$ for any action $\\Gamma\\curvearrowright \\mathcal{A}$ of a $C^*$-simple group $\\Gamma$ on a unital $C^*$-algebra $\\mathcal{A}$, and use it to prove a one-to-one correspondence between stationary states on $\\mathcal{A}$ and those on $\\Gamma\\ltimes_r \\mathcal{A}$.

2014

We present a classification theorem for a class of unital simple separable amenable Zstable C *-algebras by the Elliott invariant. This class of simple C *-algebras exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable Zstable C *-algebras. Moreover, it contains all unital simple separable amenable C *-algebras which satisfy the UCT and have finite rational tracial rank. Definition 2.8. Let A be a unital C*-algebra and let u ∈ U (A). We write Ad u for the automorphism a → u * au for all a ∈ A. Suppose B ⊆ A is a unital C*-subalgebra. Denote by Inn(B, A) the set of all those monomorphisms ϕ : B → A such that there exists a sequence of unitaries {u n } ⊂ A so that ϕ(b) = lim n→∞ u * n bu n for all b ∈ B. Definition 2.9. Denote by N the class of separable amenable C *-algebras which satisfy the Universal Coefficient Theorem (UCT). Denote by Z the Jiang-Su algebra ([44]). Note that Z has a unique trace and K i (Z) = K i (C) (i = 0, 1). A C*-algebra A is said to be Z-stable if A ∼ = A ⊗ Z. Definition 2.10. Let A be a unital C*-algebra. Recall that, following Dȃdȃrlat and Loring ([17]), one defines K(A) = i=0,1 K i (A) ⊕ i=0,1 k≥2 K i (A, Z/kZ). (e 2.1) There is a commutative C *-algebra C k such that one may identify K i (A⊗C k) with K i (A, Z/kZ). Let A be a unital separable amenable C*-algebra, and let B be a σ-unital C*-algebra. Following Rørdam ([93]), KL(A, B) is the quotient of KK(A, B) by those elements represented by limits of trivial extensions (see [61]). In the case that A satisfies the UCT, Rørdam defines KL(A, B) = KK(A, B)/P, where P is the subgroup corresponding to the pure extensions of the K * (A) by K * (B). In [17], Dȃdȃrlat and Loring proved that KL(A, B) = Hom Λ (K(A), K(B)). (e 2.2) Now suppose that A is stably finite. Denote by KK(A, B) ++ the set of those elements κ ∈ KK(A, B) such that κ(K 0 (A)\{0}) ⊂ K 0 (B) + \{0}. Suppose further that both A and B are unital. Denote by KK e (A, B) ++ the subset of those κ ∈ KK(A, B) ++ such that κ([1 A ]) = [1 B ]. Denote by KL e (A, B) ++ the image of KK e (A, B) ++ in KL(A, B). Definition 2.11. Let A and B be two C *-algebras and let ϕ : A → B be a map. We will sometime, without warning, continue to use ϕ for the induced map ϕ⊗id Mn : A⊗M n → B ⊗M n. Also ϕ ⊗ 1 Mn : A → B ⊗ M n is used for the amplification which maps a to a ⊗ 1 Mn , the diagonal element with a repeated n times. Throughout the paper, if ϕ is a homomorphism, we will use ϕ * i : K i (A) → K i (B), i = 0, 1, for the induced homomorphism. We will use [ϕ] for the element in KL(A, B) (or KK(A, B) if there is no confusion) which is induced by ϕ. Suppose that T (A) = ∅ and T (B) = ∅. Then ϕ induces an affine map ϕ T : T (B) → T (A) defined by ϕ T (τ)(a) = τ (ϕ(a)) for all τ ∈ T (B) and a ∈ A s.a.. Denote by ϕ ♯ : Aff(T (A)) → Aff(T (B)) the affine continuous map defined by ϕ ♯ (f)(τ) = f (ϕ T (τ)) for all f ∈ Aff(T (A)) and τ ∈ T (B). Definition 2.12. Let A be a unital separable amenable C *-algebra and let x ∈ A. Suppose that xx * − 1 < 1 and x * x − 1 < 1. Then x|x| −1 is a unitary. Let us use x to denote x|x| −1. Let F ⊂ A be a finite subset and ε > 0 be a positive number. We say a map L : A → B is F-ε-multiplicative if L(xy) − L(x)L(y) < ε for all x, y ∈ F. By applying Theorem 4.13 (Theorem 3.9 of [61]), there exists an integer K ≥ 1 and a unitary U ∈ P M K+1 (Q({M s(n) })P, where P = diag(1 Q({M r(n) } , 1 M K (Q({M s(n) }))), such that Ad U • (Φ 1 (f) ⊕ diag(K ψ (f), ...,ψ(f)) − (Φ 2 (f) ⊕ diag(K ψ (f), ...,ψ(f)) < ε 0 /2 (e 4.126) for all f ∈ F 0. It follows that there are unitaries {U n } ∈ ∞ n=1 M r(n)+Ks(n) such that, for all large n, Ad U n • ϕ 1,n (f) ⊕ diag(K ψ n (f), ..., ψ n (f)) − ϕ 2,n (f) ⊕ diag(K ψ n (f), ..., ψ n (f)) < ε 0 /2 (e 4.127) for all f ∈ F 0. This leads to a contradiction with (e 4.121) when we choose n with R(n) ≥ K.

Canadian Mathematical Bulletin, 2000

Let 𝒵 be the unital simple nuclear infinite dimensional C*-algebra which has the same Elliott invariant as ℂ, introduced in [9]. A C*-algebra is called 𝒵-stable if A ≅ A ⊗ 𝒵. In this note we give some necessary conditions for a unital simple C*-algebra to be 𝒵-stable.