  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 $H$.
