Residually finite-dimensional operator algebras - Clouâtre, Raphaël; Ramsey, Christopher (My University, 2019)
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 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 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 C∗-covers associated to an operator algebra.
Keywords: residual finite-dimensionality; non-selfadjoint operator algebras; tensor algebras; C⁎-covers