Couniversality for C*-algebras of residually finite-dimensional operator algebras

TL;DR

This paper investigates the limitations of the C*-envelope in capturing residual finite-dimensionality of operator algebras, revealing cases where minimal RFD C*-algebras do not exist and the space of RFD C*-algebras is not closed under certain operations.

Contribution

It constructs examples showing the failure of couniversal properties for RFD C*-algebras and analyzes the structure of RFD C*-algebras in tensor and disc algebras.

Findings

Existence of residually finite-dimensional operator algebras without minimal RFD C*-algebras

Failure of RFD C*-algebras to be closed under infima of C*-covers in tensor algebras

Specific failure example for the disc algebra

Abstract

The C*-envelope of a non self-adjoint operator algebra is known to encode many properties of the underlying subalgebra. However, the C*-envelope does not always encode the residual finite-dimensionality of an operator algebra. To elucidate this failure, we study couniversal existence in the space of residually finite-dimensional (RFD) C*-algebras attached to a fixed operator algebra. We construct several examples of residually finite-dimensional operator algebras for which there does not exist a minimal RFD C*-algebra, answering a question of the first two authors. For large swathes of tensor algebras of C*-correspondences, we also prove that the space of RFD C*-algebras fails to be closed under infima of C*-covers. In the case of the disc algebra, we are able to achieve this failure for a single pair of RFD C*-algebras.