Cohomological dimension for C*-algebras

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.<br/><br/>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.<br/><br/>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.

Name dc:type.qualificationname

Doctor of Philosophy

Level dc:type.qualificationlevel

Doctoral Thesis

Grantor dc:publisher.institution

Queen's University Belfast

Year dc:date.issued

2021

Author dc:creator

• Rosbotham, Michael

Advisor dc:contributor.advisor

• De Chiara, Gabriele

• Operator module
• C*-algebra
• exact category
• dimension
• homological algebra
• cohomology
• operator space

Language dc:language

eng

Identifier

oai:pure.qub.ac.uk/portal:studenttheses/8b5752eb-db39-425e-a216-15803fd48d4b

OAI identifier oai:identifier

oai:pure.qub.ac.uk/portal:studenttheses/8b5752eb-db39-425e-a216-15803fd48d4b