Maximal C^*-covers and residual finite-dimensionality

TL;DR

This paper investigates the structure of residually finite-dimensional operator algebras and their C*-covers, establishing conditions for the existence of maximal RFD covers and extending known characterizations to non self-adjoint cases.

Contribution

It introduces a lattice framework for C*-covers, identifies the largest RFD C*-cover for RFD operator algebras, and extends Hadwin's characterization to non self-adjoint algebras.

Findings

Equated C*-covers with a complete lattice from the spectrum of the maximal C*-cover.

Identified the largest RFD C*-cover for RFD operator algebras.

Extended Hadwin's characterization to non self-adjoint RFD C*-algebras.

Abstract

We study residually finite-dimensional (or RFD) operator algebras which may not be self-adjoint. An operator algebra may be RFD while simultaneously possessing completely isometric representations whose generating C*-algebra is not RFD. This has provided many hurdles in characterizing residual finite-dimensionality for operator algebras. To better understand the elusive behaviour, we explore the C*-covers of an operator algebra. First, we equate the collection of C*-covers with a complete lattice arising from the spectrum of the maximal C*-cover. This allows us to identify a largest RFD C*-cover whenever the underlying operator algebra is RFD. The largest RFD C*-cover is shown to be similar to the maximal C*-cover in several different facets and this provides supporting evidence to a previous query of whether an RFD operator algebra always possesses an RFD maximal C*-cover. In closing, we present a non self-adjoint version of Hadwin's characterization of separable RFD C*-algebras.