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 C * -algebras, in the nonselfadjoint 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.

Maximal C^*-covers and residual finite-dimensionality

“…Thus, this is consistent with Question 1 having an affirmative answer. For unital operator algebras, the maximal C * -cover is known to respect countable direct sum and the free product of finitely many operator algebras [6, Proposition 2.2], [11,Theorem 5.2]. We show that the RFDmaximal C * -cover also preserves these constructions (Theorems 4.10 and 4.11).…”

“…Residual finite-dimensionality for operator algebras was first studied in [11]. Therein, a complementary analysis discussing residual finite-dimensionality of C *covers was provided.…”

“…The maximal C * -cover of an operator algebra is known to preserve certain algebraic constructions. See [6, Proposition 2.2], [7, 2.4.3], [11,Theorem 5.2], [20, Theorem 4.1] for example. We provide supporting evidence for Question 1 to have an affirmative answer by showing that the RFD-maximal C * -cover preserves a few of the same algebraic constructions (Theorems 4.10 and 4.11).…”

“…We say that an operator algebra is residually finite-dimensional (or RFD) if it may be embedded into a direct product of finitedimensional C * -algebras. This concept was recently introduced in [11] and there are still only a few characterizations of RFD operator algebras (see [9], [11]).…”

“…There have been several instances for which Question 1 has been answered in the affirmative [9], [11]. Most notably, it was shown that C * max (A) is RFD whenever A is finite-dimensional.…”