Finite-Dimensional Approximations and Semigroup Coactions for Operator Algebras
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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 when residual finite-dimensionality of an operator algebra is inherited by its maximal C∗-cover, which we establish in many cases of interest.
Original language | English |
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Journal | International Mathematics Research Notices |
Volume | 2024 |
Issue number | 1 |
Pages (from-to) | 698–744 |
ISSN | 1073-7928 |
DOIs | |
Publication status | Published - 2024 |
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