A note on proper asymptotic uniqueness for semifinite factors

TL;DR

This paper investigates conditions under which certain extensions of nuclear C*-algebras into semifinite von Neumann factors are asymptotically unitarily equivalent, linking this to KK-theory invariants.

Contribution

It establishes a criterion for proper asymptotic unitary equivalence of trivial extensions in terms of KK-theory, extending understanding of extension classification.

Findings

Characterizes asymptotic unitary equivalence via KK-theory

Provides conditions for equivalence when extensions are trivial

Links extension properties to KK-theoretic invariants

Abstract

Let $\mathcal{A}$ be a separable nuclear C*-algebra, and let $\mathcal{M}$ be a semifinite von Neumann factor with separable predual. Let $\phi, \psi: \mathcal{A} \rightarrow \mathcal{M}$ be essential trivial extensions with $\phi(a) - \psi(a) \in \mathcal{K}_{\mathcal{M}}$ for all $a \in \mathcal{A}$ such that either both $\phi$ and $\psi$ (and hence $\mathcal{A}$) are unital or both $\phi$ and $\psi$ have large complement. Then $\phi$ and $\psi$ are properly asymptotically unitarily equivalent if and only if $[\phi, \psi]_{CS} = 0$ in $KK(\mathcal{A}, \mathcal{C}(S \mathcal{K}_{\mathcal{M}}))$.