Gelfand transforms and boundary representations of complete Nevanlinna--Pick quotients

Clouâtre, Raphaël; Timko, Edward J.

doi:10.48550/arxiv.1912.10331

Cited by 1 publication

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“…This theorem is applied to some uniform algebras (Corollary 5.13), to the famous Schur-Agler class of functions (Corollary 5.14) and to some algebras of multipliers on well-behaved reproducing kernel Hilbert spaces (Corollary 5.15). Taking into account Theorem 1.2, these results shed considerable light on the structure of representations of operator algebras that play an important role in multivariate operator theory; see [12,18,19,22] and the references therein.…”

Section: Introductionmentioning

confidence: 87%

Finite-dimensional approximations and semigroup coactions for operator algebras

Clouâtre¹,

Dor-On²

2021

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The residual finite-dimensionality of a C * -algebra is known to be encoded in a topological property of its space of representations, stating that finite-dimensional representations should be dense therein. We extend this paradigm to general (possibly non-self-adjoint) operator algebras. While numerous subtleties emerge in this greater generality, we exhibit novel tools for constructing finite-dimensional approximations. One such tool is a notion of a residually finite-dimensional coaction of a semigroup on an operator algebra, which allows us to construct finite-dimensional approximations for operator algebras of functions and operator algebras of semigroups. Our investigation is intimately related to the question of whether residual finite-dimensionality of an operator algebra is inherited by its maximal C * -cover, which we resolve in many cases of interest.

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Section: Introductionmentioning

confidence: 87%