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Journal of Operator Theory

Volume 77, Issue 1, Winter 2017  pp. 19-37.

Discretization of $C^*$-algebras

Authors:  Chris Heunen (1) and Manuel L. Reyes (2)
Author institution:(1) School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, U.K.
(2) Department of Mathematics, Bowdoin College, Brunswick, ME 04011--8486, U.S.A.

Summary: We investigate how a $C^*$-algebra could consist of functions on a noncommutative set: a \textit{discretization} of a $C^*$-algebra $A$ is a $*$-homomorphism $A \to M$ that factors through the canonical inclusion $C(X) \subseteq \linfty(X)$ 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 $H$.

Keywords: noncommutative topology, noncommutative set, function algebra, discrete space, profinite completion, pure state, diffuse measure, spectrum obstruction

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