Decomposable approximations and coloured isomorphisms for C*-algebras

Castillejos Lopez, Jorge (2016) Decomposable approximations and coloured isomorphisms for C*-algebras. PhD thesis, University of Glasgow.

Abstract

In this thesis we introduce nuclear dimension and compare it with a stronger form of the completely positive approximation property.
We show that the approximations forming this stronger characterisation of the completely positive approximation property witness finite nuclear dimension if and only if the underlying C*-algebra is approximately finite dimensional.
We also extend this result to nuclear dimension at most omega.

We review interactions between separably acting injective von Neumann algebras and separable nuclear C*-algebras. In particular, we discuss aspects of Connes' work and how some of his strategies have been used by C^*-algebraist to estimate the nuclear dimension of certain classes of C*-algebras.

We introduce a notion of coloured isomorphisms between separable unital C*-algebras. Under these coloured isomorphisms ideal lattices, trace spaces, commutativity, nuclearity, finite nuclear dimension and weakly pure infiniteness are preserved.
We show that these coloured isomorphisms induce isomorphisms on the classes of finite dimensional and commutative C*-algebras.
We prove that any pair of Kirchberg algebras are 2-coloured isomorphic and any pair of separable, simple, unital, finite, nuclear and Z-stable C*-algebras with unique trace which satisfy the UCT are also 2-coloured isomorphic.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Keywords: Operator algebras.
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Supervisor's Name: White, Prof. Stuart
• Date of Award: 2016
• Depositing User: Jorge Castillejos Lopez
• Unique ID: glathesis:2016-7650
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 13 Oct 2016 14:35
• Last Modified: 28 Oct 2016 07:41
• URI: https://theses.gla.ac.uk/id/eprint/7650