An invitation to stable rank for C*-algebras

23.10.2024, 10:15  –  2.09.0.14
Forschungsseminar: Gruppen und Operatoralgebren

Jamie Bell (University of Münster)

Thanks to Gelfand duality, C*-algebras are often considered noncommutative topological spaces. A very profitable approach to studying C*-algebras is to formulate noncommutative generalisations of classical topological notions. This philosophy leads, for instance, to the noncommutative geometry programme and operator K-theory.

For C*-algebras, the concept of Lebesgue covering dimension stratifies into many different notions; in any case, a low noncommutative dimension is considered an important regularity property for C*-algebras. The need for such notions is brought into sharp focus by the success of the Elliott classification programme for classifying simple, separable and nuclear C*-algebras, where finite nuclear dimension (equivalently, \(\mathcal{Z}\)-stability) is -- modulo the UCT, which may be automatic for nuclear C*-algebras -- the linchpin required to obtain a complete classification theorem.

In the early 80s, Marc Rieffel introduced the first noncommutative generalisation of covering dimension, stable rank, to answer questions related to non-stable K-theory. In the intervening years, C*-algebras with stable rank one have emerged as a rich class with many desirable properties. In this talk, we introduce stable rank for C*-algebras and discuss examples of C*-algebras having low stable rank, particularly those arising from groups and dynamical systems. This is complemented by a survey of some applications of stable rank one. Time permitting, we discuss some open problems and mention approaches to generalise stable rank one results to crossed products arising from non-amenable topological dynamical systems. This is joint work in progress with Shirly Geffen, Sven Raum and Jonathan Taylor.