The generator rank of C*-algebras

TL;DR

This paper introduces the generator rank for C*-algebras, a new invariant measuring the density of generating n-tuples, which has favorable permanence properties and applies to AF-algebras.

Contribution

It defines the generator rank as a robust invariant for C*-algebras, demonstrating its stability under various algebraic operations and providing new insights into their structure.

Findings

Every AF-algebra is generated by a single operator.

Generator rank remains stable under passing to ideals, quotients, or inductive limits.

The invariant improves understanding of generator properties in C*-algebras.

Abstract

We show that every AF-algebra is generated by a single operator. This was previously unclear, since the invariant that assigns to a C*-algebra its minimal number of generators lacks natural permanence properties. In particular, it may increase when passing to ideals or inductive limits. To obtain a better behaved theory, we not only ask if a C*-algebra is generated by $n$ elements, but also if generating $n$-tuples are dense. This defines the generator rank, which we show has many natural permanence properties: it does not increase when passing to ideals, quotients or inductive limits.