On classification of simple non-unital amenable C*-algebras, II

Abstract

We present a classification theorem for amenable simple stably projectionless C*-algebras with generalized tracial rank one whose $K_0$ vanish on traces which satisfy the Universal Coefficient Theorem. One of them is denoted by ${\mathcal{Z}}_0$ which has a unique tracial state and $K_0({\mathcal{Z}}_0)=\mathbb{Z}$ and $K_1({\mathcal{Z}}_0)=\{0\}.$ Let $A$ and $B$ be two separable simple $C^{\ast }$-algebras satisfying the UCT and have finite nuclear dimension. We show that $A\otimes {\mathcal{Z}}_0\cong B\otimes {\mathcal{Z}}_0$ if and only if $\mathrm{Ell}(B\otimes {\mathcal{Z}}_0)=\mathrm{Ell}(B\otimes {\mathcal{Z}}_0).$ A class of simple separable $C^{\ast }$-algebras which are approximately sub-homogeneous whose spectra having bounded dimension is shown to exhaust all possible Elliott invariant for $C^{\ast }$-algebras of the form $A\otimes {\mathcal{Z}}_0,$ where $A$ is any finite separable simple amenable $C^{\ast }$-algebras. Suppose that $A$ and $B$ are two finite separable simple $C^{\ast }$-algebras with finite nuclear dimension satisfying the UCT such that traces vanishe on $K_0(A)$ and $K_0(B)$ (but arbitrary $K_1$). One consequence of the main results in this situation is that $A\cong B$ if and only if $A$ and $B$ have the isomorphic Elliott invariant.

a homotopy lemma and an appendix are added. Revision of Feb. 2020

References in corpus (2)

• Approximate Homotopy of Homomorphisms from $C(X)$ into a Simple $C^{\ast }$-algebra
• A classification of inductive limits of splitting interval algebras

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