Perturbations of C*-algebraic invariants

TL;DR

This paper investigates how certain structural properties of C*-algebras, such as the similarity property and invariants like K-theory and traces, are preserved under small perturbations measured by Kadison and Kastler's metric.

Contribution

It demonstrates that the similarity property and related invariants are stable under small perturbations in the Kadison-Kastler metric, extending understanding of C*-algebra stability.

Findings

Similarity property transfers to close C*-algebras.

Closeness preserves isomorphism of Elliott invariants when the similarity property holds.

Closeness affects commutants and tensor products in predictable ways.

Abstract

Kadison and Kastler introduced a metric on the set of all C$^*$-algebras on a fixed Hilbert space. In this paper structural properties of C$^*$-algebras which are close in this metric are examined. Our main result is that the property of having a positive answer to Kadison's similarity problem transfers to close C$^*$-algebras. In establishing this result we answer questions about closeness of commutants and tensor products when one algebra satisfies the similarity property. We also examine $K$-theory and traces of close C$^*$-algebras, showing that sufficiently close algebras have isomorphic Elliott invariants when one algebra has the similarity property.