Residually finite-dimensional 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 ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-covers associated to an operator algebra.

34 pages