Asymptotic unitary equivalence and classification of simple amenable C ∗-algebras

Abstract

Let C and A be two unital separable amenable simple C * -algebras with tracial rank no more than one. Suppose that C satisfies the Universal Coefficient Theorem and suppose that ϕ 1 , ϕ 2 : C → A are two unital monomorphisms. We show that there is a continuous path ofT and a rotation related map R ϕ1,ϕ2 associated with ϕ 1 and ϕ 2 is zero.Applying this result together with a result of W. Winter, we give a classification theorem for a class A of unital separable simple amenable C * -algebras which is strictly larger than the class of separable C * -algebras whose tracial rank are zero or one. Tensor products of two C * -algebras in A are again in A. Moreover, this class is closed under inductive limits and contains all unital simple ASH-algebras whose state spaces of K 0 is the same as the tracial state spaces as well as some unital simple ASH-algebras whose K 0 -group is Z and tracial state spaces are any metrizable Choquet simplex. One consequence of the main result is that all unital simple AH-algebras which are Z-stable are isomorphic to ones with no dimension growth.

K-theoretic rigidity and slow dimension growth

“…(Corollary 1.4) Let C denote the class of unital simple ASH algebras with slow dimension growth in which projections separate traces. By Corollary 1.3, each A ∈ C is Z-stable, and so by the main result of [16] (based on [35]), we need only establish the conclusion of Corollary 1.4 for the collection C ′ consisting of algebras of the form A ⊗ U with A ∈ C and U a UHF algebra of infinite type. These algebras are Z-stable unital simple ASH algebras with real rank zero, and so the desired classification result is given by Corollary 2.5 of [32].…”

Minimal dynamics and the classification of C*-algebras

“…Theorem F. Let A, B be unital simple separable nuclear C * -algebras which absorb the Jiang-Su algebra Z tensorially (7,19). Suppose that for any prime l, the tensor products …”

Approximate unitary equivalence in simple $C^{*}$-algebras of tracial rank one

“…It is important that in 11.5 and 11.6 C * -algebra C is not assumed be simple, in particular, C could be commutative. These results will be used in subsequent papers where we study the so-called the Basic Homotopy Lemma ( [34]) and asymptotic unitary equivalence ( [35]) in simple C * -algebras with tracial rank one.…”