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Abstract: It was known from work of an Huef, Laca, and Raeburn that the right Toeplitz system for the ax+b semigroup over the naturals exhibits more than one equilibrium state at it's critical temperature, but otherwise the structure of the supercritical equilibria was unknown. In recent work with Marcelo Laca, we obtained a classification of these supercritical equilibria. Using the algebraic structure of the diagonal, we rederived a result of Afsar, Larsen, and Neshveyev which provides a correspondence between the KMS beta states and a class of measures on the circle. We provide explicit formulas for these measures and their Fourier coefficients, and describe the phase transition using properties of the Lerch zeta function.

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Abstract: ​​ I will talk about the C*-algebra generated by the right regular representation of the semigroup of affine transformations of the natural numbers. This `right Toeplitz algebra' has three distinguished ideals that give three boundary quotients, and I will describe their structure and discuss their representations. From this we can see that the right Toeplitz algebra is very different from the left Toeplitz algebra studied by Raeburn and myself, although Cuntz-Echterhoff-Li (2013) have shown they have the same K-theory. When we compute KMS states, we see that the low temperature equilibrium states of the right Toeplitz algebra are also parametrized by measures on the​ circle, just as Raeburn and I showed for the left one. So right and left Toeplitz algebras also share the same crystalline phases. This is joint work with Astrid an Huef and Iain Raeburn [NZJM 2021].

Abstract: We prove that the class of crossed product C*-algebras arising from the action of the multiplicative group of a number field on its rings of finite adeles is rigid in the following explicit sense: Any *-isomorphism between full corners of such C*-algebras gives rise to an isomorphism between the underlying number fields that is constructed from the *-isomorphism of corners. I will discuss this result, an application, and I will indicate the idea of proof if time permits. This talk is based on joint work with Takuya Takeishi (Kyoto Institute of Technology).

Abstract: We consider a locally compact Hausdorff etale groupoid G constructed from a family of k commuting local homemorphisms acting on a compact Hausdorff space X. We characterize the continuous one-cocycles in the groupoid G taking on real values in terms of k-tuples of continuous real-valued on functions on X satisfying certain canonical identities. Under appropriate conditions, we construct a continuous one-parameter automorphism group acting on the C*-algebra associated to G coming from a continuous real-valued one-cocycle on G. The question of the existence of KMS states on the groupoid C*-algebra associated to these automorphism groups is addressed. The work discussed is joint with C. Farsi, L. Huang, and A. Kumjian.

Abstract: I will report on progress on the problem of constructing an example of what Connes' calls a `twisted' (finitely summable) spectral triple over the crossed product of a (classical) hyperbolic group acting on its boundary. This particular triple is important because it represents the Dirac class of the action, and is the analogue in Type III of Connes' Dolbeault spectral triple over the irrational torus. (The Dirac class equals in this case the K-homology class of the boundary extension of the group, as well.) My description uses Patterson-Sullivan theory fairly extensively.

Abstract: I will report on progress on the problem of constructing an example of what Connes' calls a `twisted' (finitely summable) spectral triple over the crossed product of a (classical) hyperbolic group acting on its boundary. This particular triple is important because it represents the Dirac class of the action, and is the analogue in Type III of Connes' Dolbeault spectral triple over the irrational torus. (The Dirac class equals in this case the K-homology class of the boundary extension of the group, as well.)

My description uses Patterson-Sullivan theory fairly extensively.

Abstract: I will report on progress on the problem of constructing an example of what Connes' calls a `twisted' (finitely summable) spectral triple over the crossed product of a (classical) hyperbolic group acting on its boundary. This particular triple is important because it represents the Dirac class of the action, and is the analogue in Type III of Connes' Dolbeault spectral triple over the irrational torus. (The Dirac class equals in this case the K-homology class of the boundary extension of the group, as well.) My description uses Patterson-Sullivan theory fairly extensively.

Zoom link.

Abstract: I will report on progress on the problem of constructing an example of what Connes' calls a `twisted' (finitely summable) spectral triple over the crossed product of a (classical) hyperbolic group acting on its boundary. This particular triple is important because it represents the Dirac class of the action, and is the analogue in Type III of Connes' Dolbeault spectral triple over the irrational torus. (The Dirac class equals in this case the K-homology class of the boundary extension of the group, as well.) My description uses Patterson-Sullivan theory fairly extensively. I will explain this theory for the hyperbolic plane and free groups, in the first talk, and define the `Dirac operator' in the second.

Abstract: I will review the definition of the universal Toeplitz C*-algebra Tu(P) associated to a submonoid P of a group. Tu(P) is given by a presentation based on Xin Li’s semilattice of constructible ideals with an extra condition. After briefly mentioning faithful representations and a uniqueness theorem for the reduced Toeplitz algebra Tλ(P), I will concentrate on the associated universal boundary quotient ∂Tu(P) and its reduced version ∂Tλ(P), and give sufficient conditions for the latter to be purely infinite simple. I intend to discuss some new applications, mainly to monoids arising from non-maximal orders in number fields but also to right LCM monoids with nontrivial units. This is joint work with Camila F. Sehnem.