Classification of simple C*-algebras of tracial topological rank zero

Abstract

We give a classification theorem for unital separable simple nuclear C * -algebras with tracial topological rank zero which satisfy the Universal Coefficient Theorem. We prove that if A and B are two such C * -algebras and

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

“…for x ∈ X j and j = 1, 2, ..., L. Moreover, as in 3.4 of [26], r(j) ≥ 2 (r 1 −r)D . There is a projection…”

“…As in 6.2 of [26], there exists a finite subset G and a small δ > 0 such that a δ-G-multiplicative contractive completely positive linear map L induces a homomorphism…”

“…Note that, since A is simple, by Proposition 3.4 of [26], we may assume that r(j) ≥ 8/(r 1 − r)σ j = 1, 2, ..., m.…”

“…Proof. It follows from 10.9 of [26] (by applying a theorem of Villadsen [47]) that there exists a unital simple AH-algebra B with property (J) such that…”

A simple separable C*-algebra not isomorphic to its opposite algebra

“…The recent work on classification of simple nuclear C*-algebras, for example [13] and [10] in the purely infinite case and [7] and [11] in the stably finite case, suggests that all simple nuclear C*-algebras might be isomorphic to their opposites.…”