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arXiv · 1607.03376

Discretization of C*-algebras

Chris Heunen·Manuel L. Reyes
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Abstract

We investigate how a C*-algebra could consist of functions on a noncommutative set: a discretization of a C*-algebra $A$ is a $*$-homomorphism $A \to M$ that factors through the canonical inclusion $C(X) \subseteq \ell^\infty(X)$ when restricted to a commutative C*-subalgebra. Any C*-algebra admits an injective but nonfunctorial discretization, as well as a possibly noninjective functorial discretization, where $M$ is a C*-algebra. Any subhomogenous C*-algebra admits an injective functorial discretization, where $M$ is a W*-algebra. However, any functorial discretization, where $M$ is an AW*-algebra, must trivialize $A = B(H)$ for any infinite-dimensional Hilbert space $H$.

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BibTeXRIS

Chris Heunen, Manuel L. Reyes. 2017-02-15. Discretization of C*-algebras. https://doi.org/10.7900/jot.2015jun16.2109

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