Topological invariants for bundles of C∗-algebras with symmetries

Bianchi, Valerio 2025. Topological invariants for bundles of C∗-algebras with symmetries. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

By a classical result of Dixmier and Douady, one can classify locally trivial bundles of C ∗ -algebras with the compact operators as fibres by methods from algebraic topology. Building on work by Dadarlat, Evans and Pennig in this thesis we develop an equivariant generalisation of this theory to a wider class of C ∗ -algebras starting with a concrete setting. More precisely, we consider infinite tensor product actions of G = Z/pZ on the UHF-algebra D = End (V ) ⊗∞ for a finite-dimensional unitary G-representation V and determine the equivariant homotopy type of the group Aut(D ⊗ K), where K are the compact operators on ℓ 2 (G) ⊗ H0 for a separable Hilbert space H0 with dim(H0) = ∞. We show that this group carries an equivariant infinite loop space structure revealing it as the first space of a G-spectrum, which we prove to be equivalent to the positive units gl1(KU D)+ of equivariant KU D-theory. Here, KU D is a G-spectrum representing X 7→ KG ∗ (C(X)⊗D). As a consequence we are able to classify equivariant (D⊗K)-bundles over finite CW complexes by the first group of the cohomology theory associated to gl1(KU D)+. In doing so, we propose an equivariant infinite loop spaces machine which may be of independent interest.

Item Type:	Thesis (PhD)
Date Type:	Completion
Status:	Unpublished
Schools:	Schools > Mathematics
Uncontrolled Keywords:	1. K-theory 2. Operator algebras 3. Equivariant stable homotopy theory 4. Spectra of units 5. Dixmier-Douady 6. Bundles of C*-algebras
Date of First Compliant Deposit:	19 February 2026
Last Modified:	19 Feb 2026 15:51
URI:	https://orca.cardiff.ac.uk/id/eprint/185066

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