## Residual finite-dimensionality and realizations of approximate rigidity phenomena for operator algebras

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##### Date

2024-07-11

##### Authors

Thompson, Ian

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##### Abstract

Representations on a Hilbert space are a common tool for understanding both C*-algebras and (non self-adjoint) operator algebras. For operator algebras, a major theme has been to capitalize on the structural theory of an enveloping C*-algebra (or C*-cover) to better understand the underlying subalgebra. This philosophy has spanned several advancements, yet there is a subtlety involved in the process: the C*-algebra can vary greatly depending on the choice of representation of the operator algebra. Thus, it becomes advantageous to leverage information from several C*-covers to better understand the underlying operator algebra. Here, we focus on interactions between an operator algebra and its C*-covers, with an emphasis on sufficiency and maximality conditions for two fundamental classes of representations: the finite-dimensional representations and the non-commutative Choquet boundary, respectively. Both families offer separate advantages because the former class is more tractable, yet the latter induces a minimal representation due to work of Arveson.
In Chapter 3, we analyze the residual finite-dimensionality of the maximal C*-cover of an operator algebra. To this end, we consider several C*-covers that are formed through families of finite-dimensional representations and compare these C*-covers with the maximal C*-cover. Along the way, we generalize Hadwin's characterization of separable residually finite-dimensional C*-algebras.
In Chapter 4, we study maximality conditions on the non-commutative Choquet boundary. A conjecture of Arveson asserts that maximality conditions imply a rigidity property that is, a priori, much stronger. In this chapter, we uncover a significant localization procedure that generalizes several past attempts at Arveson's conjecture. This localization procedure is also applied to another conjecture of Arveson that concerns quotient modules of the Drury-Arveson space.
Arveson's rigidity conjecture is originally inspired by a development in approximation theory due to Šaškin. In Chapter 5, we achieve one non-commutative analogue to Šaškin's theorem. In the setting of classical function theory, this encodes a maximality condition for the Choquet boundary with a rigidity property. We find that a similar phenomenon is still true for a large class of C*-algebras.

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##### Keywords

Residually finite-dimensional, Korovkin approximation, Operator algebra, Non self-adjoint, C*-algebra, Operator space, Hyperrigidity, Approximate unitary equivalence