Discretization of C*-algebras

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 → M that factors through the canonical inclusion C(X) ⊆ ℓ∞;(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