Year
2023
Degree Name
Doctor of Philosophy
Department
School of Mathematics and Statistics
Abstract
We introduce a generalisation of Cartan pairs of C∗-algebras, which we call Cartan triples, consisting of a C∗-algebra A, a distinguished abelian subalgebra D, and the relative commutant D′ of D in A. From a Cartan triple (A,D′,D) we construct a Fell bundle E over an effective groupoid G, such that the fibres over units in the groupoid are unital C∗-algebras. We show that A is isomorphic to C∗ r (E;G), via an isomorphism which sends D to the canonical commutative subalgebra of C∗r (E;G), consisting of sections taking values in the scalar multiples of identity elements in the fibres over units. We introduce the integrated bundle of a Fell bundle B over a second countable, locally compact, Hausdorff, amenable, groupoid G with Iso◦(G) closed. The integrated bundle B is a Fell bundle over the quotient of G by Iso◦(G), which is an effective groupoid. We show that the C∗-algebra of the integrated bundle recovers the C∗-algebra of the original bundle. Furthermore, after associating a canonical Cartan triple to the C∗-algebra of a twist over an appropriate groupoid, we show that the Fell bundle constructed from this triple is isomorphic to the integrated bundle of the line bundle of the twist.
Recommended Citation
Zimmerman, Joel Alexander, Cartan Triples and Fell Bundle Models for C*-algebras, Doctor of Philosophy thesis, School of Mathematics and Statistics, University of Wollongong, 2023. https://ro.uow.edu.au/theses1/1705
FoR codes (2008)
010108 Operator Algebras and Functional Analysis
Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.