Maximal C^*-covers and residual finite-dimensionality

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.

Quasitriangular operator algebras

“…[14], [15] for alternative approaches). Note that an operator algebra might be RFD while its enveloping C*-algebra is not so [14,Example 3.4] and [53,Example 1].…”