06.11.2024, 14:00 - 16:00
– Campus Golm, Building 9, Room 2.22 and via Zoom
Institutskolloquium
Graphon Models for Inhomogeneous Random Graphs
Olga Klopp (Paris), Nicolas Verzelen (Montpellier)
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.