Discretization of C*-algebras

TL;DR

This paper explores how C*-algebras can be represented as functions on noncommutative sets through discretizations, analyzing their properties and limitations, especially in the context of functoriality and algebraic structures.

Contribution

It introduces and compares various discretization methods for C*-algebras, highlighting their existence, injectivity, functoriality, and limitations across different algebra classes.

Findings

Injective nonfunctorial discretization exists for all C*-algebras.

Subhomogeneous C*-algebras admit injective functorial discretizations.

Functorial discretizations with AW*-algebras trivialize infinite-dimensional B(H).

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 $H$.