# 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 $H$.

**DOI: **http://dx.doi.org/10.7900/jot.2015jun16.2109

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

Contents
Full-Text PDF