Unitary equivalences for essential extensions of $C^*$-algebras

TL;DR

This paper investigates the classification of essential extensions of unital separable C*-algebras by analyzing their unitary equivalences and establishing a bijection with certain K-theoretic groups, especially in the context of approximate unitary equivalence.

Contribution

It introduces a classification framework linking strong unitary equivalence classes of extensions to quotient groups of K_0(B), and explores approximate equivalence when B is simple with continuous scale.

Findings

A bijection between a quotient of K_0(B) and strong unitary equivalence classes.

When the K_0(B) quotient is zero, full extensions are strongly unitarily equivalent.

Characterization of approximate unitary equivalence in simple C*-algebras with continuous scale.

Abstract

Let $A$ be a unital separable \CA and $B=C\otimes {\cal K},$ where $C$ is a unital \CA. Let $\tau: A\to M(B)/B$ be a weakly unital full essential extensions of $A$ by $B.$ We show that there is a bijection between a quotient group of $K_0(B)$ onto the set of strong unitary equivalence classes of weakly unital full essential extensions $\sigma$ such that $[\sigma]=[\tau]$ in $KK^1(A, B).$ Consequently, when this group is zero, unitarily equivalent full essential extensions are strongly unitarily equivalent. When $B$ is a non-unital but $\sigma$-unital simple \CA with continuous scale, we also study the problem when two approximately unitarily equivalent essential extensions are strongly approximately unitarily equivalent. A group is used to compute the strongly approximate unitary equivalence classes in the same approximate unitary equivalent class of essential