Discretization of $C^*$-algebras

Chris Heunen,Manuel L Reyes

doi:10.7900/jot.2015jun16.2109

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 $H$.

Highlights

In operator algebra it is common practice to think of a C*-algebra as representing a noncommutative analogue of a topological space, and to think of a W*-algebra as representing a noncommutative analogue of a measurable space
What would it mean to make precise the notion of a C*-algebra A as a ‘noncommutative ring of continuous functions’ ? The present article explores the idea that one should first embed A in an appropriate noncommutative algebra of ‘bounded functions on the underlying quantum set of the spectrum of A’, just like any topological space embeds in a discrete one [1, 4]
It is tempting to demand that such a ‘noncommutative function ring’ be an atomic W*-algebra, but we work more generally under the mere assumption that they be AW*-algebras

Summary

Introduction

In operator algebra it is common practice to think of a C*-algebra as representing a noncommutative analogue of a topological space, and to think of a W*-algebra as representing a noncommutative analogue of a measurable space. A discretization of a unital C*-algebra A is a unital ∗-homomorphism φ : A → M to an AW*-algebra M whose restriction to each commutative unital C*subalgebra C ∼= C(X) factors through the natural inclusion C(X) → l∞(X) via a morphism l∞(X) M in AWstar, so that the following diagram commutes.

Results

Conclusion