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arXiv · 2107.14275

Maximal C^*-covers and residual finite-dimensionality

Ian Thompson
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Abstract

We study residually finite-dimensional (or RFD) operator algebras which may not be self-adjoint. An operator algebra may be RFD while simultaneously possessing completely isometric representations whose generating C*-algebra is not RFD. This has provided many hurdles in characterizing residual finite-dimensionality for operator algebras. To better understand the elusive behaviour, we explore the C*-covers of an operator algebra. First, we equate the collection of C*-covers with a complete lattice arising from the spectrum of the maximal C*-cover. This allows us to identify a largest RFD C*-cover whenever the underlying operator algebra is RFD. The largest RFD C*-cover is shown to be similar to the maximal C*-cover in several different facets and this provides supporting evidence to a previous query of whether an RFD operator algebra always possesses an RFD maximal C*-cover. In closing, we present a non self-adjoint version of Hadwin's characterization of separable RFD C*-algebras.

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Ian Thompson. 2021-07-29. Maximal C^*-covers and residual finite-dimensionality. https://arxiv.org/abs/2107.14275

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