Extensions of C*-algebras

TL;DR

This paper classifies essential extensions of certain C*-algebras using KK theory, characterizes liftability and equivalence relations, and extends the Voiculescu theorem to new classes of C*-algebras.

Contribution

It provides a KK-theoretic classification of essential extensions of separable amenable C*-algebras by simple C*-algebras with continuous scale, including liftability criteria and a noncommutative Weyl--von Neumann theorem.

Findings

Classification of extensions via KK theory.

Criteria for liftability of extensions.

A noncommutative Weyl--von Neumann theorem for certain C*-algebras.

Abstract

Let $A$ be a separable amenable $C^*$-algebra and $B$ a non-unital and $\sigma$-unital simple $C^*$-algebra with continuous scale ($B$ need not be stable). We classify, up to unitary equivalence, all essential extensions of the form $0 \rightarrow B \rightarrow D \rightarrow A \rightarrow 0$ using KK theory. There are characterizations of when the relation of weak unitary equivalence is the same as the relation of unitary equivalence, and characterizations of when an extension is liftable (a.k.a.~trivial or split). In the case where $B$ is purely infinite, an essential extension $\rho : A \rightarrow M(B)/B$ is liftable if and only if $[\rho]=0$ in $KK(A, M(B)/B)$. When $B$ is stably finite, the extension $\rho$ is often not liftable when $[\rho]=0$ in $KK(A, M(B)/B).$ Finally, when $B$ additionally has tracial rank zero and when $A$ belongs to a sufficiently regular class of unital separable amenable $C^*$-algebras, we have a version of the Voiculescu noncommutative Weyl--von Neumann theorem: Suppose that $\Phi, \Psi: A \rightarrow M(B)$ are unital injective homomorphisms such that $\Phi(A) \cap B = \Psi(A) \cap B = \{ 0 \}$ and $\tau \circ \Phi = \tau \circ \Psi$ for all $\tau \in T(B),$ {the tracial state space of $B.$} Then there exists a sequence $\{ u_n \}$ of unitaries in $M(B)$ such that (i) $u_n \Phi(a) u_n^* - \Psi(a) \in B$ for all $a \in A$ and $n \geq 1$, (ii) $\| u_n \Phi(a) u_n^* - \Psi(a) \| \rightarrow 0$ as $n \rightarrow \infty$ for all $a \in A$.