C*-algebras nearly contained in type I algebras

TL;DR

This paper investigates near inclusions of C*-algebras, demonstrating that under certain conditions, a C*-algebra close to a type I algebra inherits the type I property and can be embedded into it.

Contribution

It establishes that near inclusions preserve the type I property for C*-algebras satisfying Kadison's similarity problem and provides an embedding result.

Findings

A near inclusion of a C*-algebra into a separable type I algebra implies the algebra is also type I.

C*-algebras satisfying Kadison's similarity problem can be embedded into type I algebras under near inclusion.

The paper extends understanding of structural stability of C*-algebras under approximate inclusions.

Abstract

In this paper we consider near inclusions $A\subseteq_\gamma B$ of C$^*$-algebras. We show that if $B$ is a separable type I C*-algebra and $A$ satisfies Kadison's similarity problem, then $A$ is also type I and use this to obtain an embedding of $A$ into $B$.