Finite dimensional approximations in operator algebras

TL;DR

This paper characterizes residually finite dimensional (RFD) operator algebras via their matrix state space and approximation properties, extending known results from $C^*$-algebras to non-self-adjoint cases.

Contribution

It provides a characterization of RFD operator algebras in terms of matrix states and shows the equivalence with approximation by finite-dimensional representations, including a counterexample.

Findings

RFD operator algebras are characterized by their matrix state space.

Every representation of an RFD operator algebra can be approximated by finite-dimensional ones.

An example operator algebra where approximation in the strong operator topology fails.

Abstract

A non-self-adjoint operator algebra is said to be residually finite dimensional (RFD) if it embeds into a product of matrix algebras. We characterize RFD operator algebras in terms of their matrix state space, and moreover show that an operator algebra is RFD if and only if every representation can be approximated by finite dimensional ones in the point weak operator topology. This is a non-self-adjoint version of a theorem of Exel and Loring for $C^*$-algebras. Moreover, we construct an example of an operator algebra for which approximation in the point strong operator topology is not possible. As a consequence, the maximal $C^*$-algebra generated by this operator algebra is not RFD. This answers questions of Clou\^atre and Ramsey and of Clou\^atre and Dor-On.