Perturbations of intermediate C*-subalgebras for simple C*-algebras

TL;DR

This paper investigates how small perturbations affect intermediate C*-subalgebras within simple C*-algebras, showing that close subalgebras are unitarily equivalent under certain conditions, with implications for their classification.

Contribution

It establishes conditions under which intermediate C*-subalgebras are unitarily equivalent when perturbed slightly, even without nuclearity, and relates this to the structure of the relative commutant.

Findings

Close simple intermediate C*-subalgebras are unitarily equivalent.

The unitary implementing equivalence can be found in the relative commutant.

If the relative commutant is trivial, the set of intermediate subalgebras is finite.

Abstract

We study uniform perturbations of intermediate C*-subalgebras of inclusions of simple C*-algebras. If a unital simple C*-algebra has a simple C*-subalgebra of finite index, then sufficiently close simple intermediate C*-subalgebras are unitarily equivalent. These C*-subalgebras need not to be nuclear. The unitary can be chosen in the relative commutant algebra. An imediate corollary is the following: If the relative commutant is trivial, then the set of intermediate C*-subagebras is a finite set.