Discretization of C*-algebras

Abstract

We investigate how a C*-algebra could consist of functions on a noncommutative set: a discretization of a C*-algebra $A$ is a $\ast$-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 $H$.

16 pages. This paper supersedes arXiv:1412.1721

References in corpus (2)

• Extending obstructions to noncommutative functorial spectra
• Dye's Theorem and Gleason's Theorem for AW*-algebras

Cited by in corpus (1)

• The finite dual coalgebra as a quantization of the maximal spectrum