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 behavior, 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.