Operator spaces and residually finite-dimensional $C^\ast$-algebras

TL;DR

This paper proves that for any operator space, the universal $C^*$-algebra containing it is residually finite-dimensional, extending previous results and providing a concise proof using a criterion by Exel and Loring.

Contribution

It extends the class of operator spaces whose universal $C^*$-algebras are residually finite-dimensional and offers a simplified proof of this property.

Findings

Universal $C^*$-algebra of any operator space is residually finite-dimensional.

The free $C^*$-algebra on any normed space is residually finite-dimensional.

Provides a concise proof based on Exel and Loring's criterion.

Abstract

For every operator space $X$ the $C^\ast$-algebra containing it in a universal way is residually finite-dimensional (that is, has a separating family of finite-dimensional representations). In particular, the free $C^\ast$-algebra on any normed space so is. This is an extension of an earlier result by Goodearl and Menal, and our short proof is based on a criterion due to Exel and Loring.