Nonamenable simple C*-algebras with tracial approximation

摘要

We construct two types of unital separable simple C*-algebras: A(Z)(C1) and A(Z)(C2), one exact but not amenable, the other nonexact. Both have the same Elliott invariant as the Jiang-Su algebra - namely, A(Z)(Ci) has a unique tracial state, @@@ (K-0(A(z)(Ci)), K-0(A(zCi))(+), [1A(z)ci]) = (Z, Z+, 1), @@@ and K-1(A(z)(Ci)) = {0} (i = 1, 2). We show that A(z)(Ci) (i = 1, 2) is essentially tracially in the class of separable L-stable C*-algebras of nuclear dimension 1. A(z)(Ci) has stable rank one, strict comparison for positive elements and no 2-quasitrace other than the unique tracial state. We also produce models of unital separable simple nonexact (exact but not nuclear) C*-algebras which are essentially tracially in the class of simple separable nuclear L-stable C*-algebras, and the models exhaust all possible weakly unperforated Elliott invariants. We also discuss some basic properties of essential tracial approximation.

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