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Abstract: We consider monoids M_k generated by the first k+1 generators of the infinite presentation of the Thompson group F

F=⟨x0,x1,x2,…∣ xjxi=xixj+1 for j>i⟩.

The standard normal form for F breaks down for these monoids and we develop a new reduced form that works in F and every Mk and allows us to analyze their constructible right ideal structure. We show that there exist embeddings Mk↪Mk+1 for which the associated Toeplitz algebras are functorial, and then we study the directed system of Toeplitz algebras λ(Mk)↪λ(Mk+1). We characterize faithful representations and show that the boundary quotients are purely infinite simple. This is joint work in progress with A. an Huef, B. Nucinkis, I. Raeburn and C. Sehnem.

Abstract: We consider monoids M_k generated by the first k+1 generators of the infinite presentation of the Thompson group F F=⟨x0,x1,x2,…∣ xjxi=xixj+1 for j>i⟩. The standard normal form for F breaks down for these monoids and we develop a new reduced form that works in F and every Mk and allows us to analyze their constructible right ideal structure. We show that there exist embeddings Mk↪Mk+1 for which the associated Toeplitz algebras are functorial, and then we study the directed system of Toeplitz algebras λ(Mk)↪λ(Mk+1). We characterize faithful representations and show that the boundary quotients are purely infinite simple. This is joint work in progress with A. an Huef, B. Nucinkis, I. Raeburn and C. Sehnem.

Abstract: I discuss a modest framework for doing Noncommutative Geometry for certain examples of `noncommutative manifolds,' namely crossed products of discrete groups acting smoothly on manifolds. The analogue of the `fundamental class' in topology for these C*-algebras is something I call a `fibre class.' I will discuss the numerous examples of these fibre classes, and explain why the definition is so great.

Abstract: I will begin with some basics about Approximately Finite dimensional C*-algebras or AF-algebras. These have played a fundamental role in many aspects of the theory and I will mention some special classes of interest. I will briefly describe the Oseledets or Multiplicative Ergodic Theorem (warning! I am not an expert) and how this can be used to create a broad class of AF-algebras with interesting features.