Residually finite-dimensional operator algebras

TL;DR

This paper explores non-selfadjoint operator algebras that are fully characterized by their finite-dimensional representations, revealing unique properties and conditions for residual finite-dimensionality in this context.

Contribution

It introduces a non-selfadjoint notion of residual finite-dimensionality and identifies conditions under which tensor algebras of C*-correspondences possess this property.

Findings

Established sufficient conditions for residual finite-dimensionality of tensor algebras

Analyzed the residual finite-dimensionality of minimal and maximal C*-covers

Highlighted differences between self-adjoint and non-selfadjoint operator algebras

Abstract

We study non-selfadjoint operator algebras that can be entirely understood via their finite-dimensional representations. In contrast with the elementary matricial description of finite-dimensional $\mathrm{C}^*$-algebras, in the non-selfadjoint setting we show that an additional level of flexibility must be allowed. Motivated by this peculiarity, we consider a natural non-selfadjoint notion of residual finite-dimensionality. We identify sufficient conditions for the tensor algebra of a $\mathrm{C}^*$-correspondence to enjoy this property. To clarify the connection with the usual self-adjoint notion, we investigate the residual finite-dimensionality of the minimal and maximal $\mathrm{C}^*$-covers associated to an operator algebra.