## Departmental PhD Thesis Exam – James Lutley

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Tuesday, June 13, 2017
3:10 p.m.
BA6183

PhD Candidate:  James Lutley
Supervisor:  Georges Elliott
Thesis title: The Structure of Diagonally Constructed ASH Algebras

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Abstract:

We introduce a class of recursive subhomogeneous algebras which are constructed using a type of diagonal map similar to those previously defined for homogeneous algebras. We call these diagonal subhomogeneous (DSH) algebras.
Using homomorphisms that also exhibit a kind of diagonal structure, we study certain limits of DSH algebras. Our first result is that a simple limit of DSH algebras with diagonal maps has stable rank one.  As an application we show that whenever $X$ is a compact Hausdorff space and $\sigma$ is a minimal homeomorphism thereof, the crossed product algebra $C^*(\mathbb{Z},X,\sigma)$ has stable rank one. We also define mean dimension in the context of these limits. Our second result is that mean dimension zero implies $\mathcal{Z}$-stability for simple separable limits of DSH algebras with diagonal maps. We also show that the tensor product of any two simple separable limit algebras of this kind is $\mathcal{Z}$-stable.

A copy of the thesis can be found here: Lutley-thesis