The topological dimension of type I C*-algebras

TL;DR

This paper introduces a noncommutative dimension theory for type I C*-algebras, showing that the topological dimension by Brown and Pedersen fits this framework and providing related rank estimates.

Contribution

It establishes the topological dimension as a noncommutative dimension theory for type I C*-algebras, expanding the understanding of dimension concepts in noncommutative topology.

Findings

Topological dimension satisfies axioms of a noncommutative dimension theory.

Provides estimates of real and stable rank based on topological dimension.

Shows the compatibility of Brown and Pedersen's topological dimension with other noncommutative dimensions.

Abstract

While there is only one natural dimension concept for separable, metric spaces, the theory of dimension in noncommutative topology ramifies into different important concepts. To accommodate this, we introduce the abstract notion of a noncommutative dimension theory by proposing a natural set of axioms. These axioms are inspired by properties of commutative dimension theory, and they are for instance satisfied by the real and stable rank, the decomposition rank and the nuclear dimension. We add another theory to this list by showing that the topological dimension, as introduced by Brown and Pedersen, is a noncommutative dimension theory of type I C*-algebras. We also give estimates of the real and stable rank of a type I C*-algebra in terms of its topological dimension.