Ulam stability for some classes of C*-algebras

TL;DR

This paper establishes stability results for finite-dimensional C*-algebras, showing that approximately *-homomorphic maps can be approximated by true *-homomorphisms with bounds independent of the specific algebras.

Contribution

It proves that approximate *-homomorphisms from finite-dimensional C*-algebras to any C*-algebra can be closely approximated by actual *-homomorphisms, with bounds depending only on the approximation quality.

Findings

Approximate *-homomorphisms are close to true *-homomorphisms.

The stability bounds depend only on the approximation error.

Results apply universally to finite-dimensional C*-algebras.

Abstract

We prove some stability results for certain classes of C*-algebras. We prove that whenever $A$ is a finite-dimensional C*-algebra, $B$ is a C*-algebra and $\phi\colon A\to B$ is approximately a $^*$-homomorphism then there is an actual $^*$-homomorphism close to $\phi$ by a factor depending only on how far is $\phi$ from being a $^*$-homomorphism and not on $A$ or $B$.