Cohomological dimension for C*-algebras

Abstract

We define and explore invariants for C*-algebras that arise as cohomological dimensions for associated categories of operator space modules. The setting of exact categories provides us with a robust framework to utilise homological techniques.

We develop initial global dimension theorems for two of these categories. In the additive category of operator modules over a C*-algebra, equipped with the exact structure of all kernel-cokernel pairs, we show how an extension theorem of Wittstock and a representation theorem of Christensen-Effros-Sinclair can be used to build injective resolutions. From there, we establish a lower bound for the associated global dimension.

We also investigate the sub–exact structure of kernel-cokernel pairs that split as completely bounded linear maps. This provides a new context in which to discuss relative homological algebra for operator modules over a C*-algebra. We provide a proof that the cohomological dimension of this exact category is zero ifand only if the C*-algebra is classically semisimple.

Date of Award	Dec 2021
Original language	English
Awarding Institution	Queen's University Belfast
Sponsors	Northern Ireland Department for the Economy
Supervisor	Martin Mathieu (Supervisor) & Gabriele De Chiara (Supervisor)