C2

Please attend if you would like to give a talk in the Logic Advanced Class this term.

Associated to every shift space, the Cuntz-Krieger algebra (C-K algebra for abbreviation) is an invariant of conjugacy defined and developed by K. Matsumoto, S. Eilers, T. Carlsen, and many of their collaborators in the last decade. In particular, Carlsen defined the C-K algebra to be the full groupoid C*-algebra of the “cover”, which is a topological system consisting of a surjective local homeomorphism on a zero-dimensional space induced by the shift space. 

In 2022, K. Brix proved that the C-K algebra of the Sturmian shift has finite nuclear dimension, where the Sturmian shift is the (unique) minimal shift space with the smallest complexity function: p_X(n)=n+1. In recent results (joint with Z. He), we show that for any minimal shift space with finitely many left special elements, its C-K algebra always have finite nuclear dimension. In fact, this can be further applied to the class of aperiodic shift spaces with non-superlinear growth complexity. 

Product systems represent powerful contemporary tools in the study of mathematical structures. A major success in the theory came from Katsura (2007), who provided a complete description of the gauge-invariant ideals of many important C*-algebras arising from product systems over Z+. This result recaptures existing results from the literature, illustrating the versatility of product system theory. The question now becomes whether or not Katsura's result can be bolstered to product systems over semigroups other than Z+ and, if so, what applications do we obtain? An answer has been elusive, owing to the more pathological nature of product systems over general semigroups. However, recent strides by Dor-On and Kakariadis (2018) supply a more tractable subclass of product systems that still includes the important cases of C*-dynamics, row-finite higher-rank graphs, and regular product systems. 

In this talk we will build a parametrisation of the gauge-invariant ideals, starting from first principles and gradually increasing in complexity. We will pay particular attention to the higher-rank subtleties that are not witnessed in Katsura's theorem, and comment on the applications.
 

A group is called W*-superrigid if its group von Neumann algebra completely remembers the original group. In this talk, I will present a recent joint work with Stefaan Vaes in which we generalize W*-superrigidity for groups in two directions. Firstly, we find a class of groups for which W*-superrigidity holds in the presence of a twist by an arbitrary 2-cocycle: the twisted group von Neumann algebra completely remembers both the original group and the 2-cocycle. Secondly, for the same class of groups, the superrigidity also holds up to virtual isomorphism.

A central theme in the 40-year-old Ribe program is the quest for metric invariants that characterize local properties of Banach spaces. These invariants are usually closely related to the geometry of certain sequences of finite graphs (Hamming cubes, binary trees, diamond graphs...) and provide quantitative bounds on the bi-Lipschitz distortion of those graphs.

A more recent program, deeply influenced by the late Nigel Kalton, has a similar goal but for asymptotic properties instead. In this talk, we will motivate the (asymptotic) notions of infrasup umbel convexity (introduced in collaboration with Chris Gartland (UC San Diego)) and bicone convexity. These asymptotic notions are inspired by the profound work of Lee, Mendel, Naor, and Peres on the (local) notion of Markov convexity and of Eskenazis, Mendel, and Naor on the (local) notion of diamond convexity. 

All these metric invariants share the common feature of being derived from point-configuration inequalities which generalize curvature inequalities.

If time permits we will discuss the values of these invariants for Heisenberg groups.

Recently, Elliott, Li, and Niu proved a classification theorem for Villadsen-type algebras using the combination of the Elliott invariant and the radius of comparison, an invariant that was introduced by Toms in order to distinguish between certain non-isomorphic AH algebras with the same Elliott invariant. This might have raised the prospect that the Elliott classification program can be extended beyond the Z-stable case by adding the radius of comparison to the invariant. I will discuss a recent preprint in which we show that this is not the case: we construct an uncountable family of nonisomorphic AH algebras with the same Elliott and same radius of comparison. We can distinguish between them using a finer invariant, which we call the local radius of comparison. This is joint work with N. Christopher Phillips.

A $C^*$-diagonal is a certain commutative subalgebra of a $C^∗$ -algebra with a rich structure. Renault and Kumjian showed that finding a $C^*$ -diagonal of a $C^∗$-algebra is equivalent to realizing the $C^*$-algebra via a groupoid. This establishes a close connection between $C^∗$-diagonals and dynamics and allows one to relate the geometric properties of groupoids to the properties of $C^∗$ -diagonals. 

In this talk, I will explore the unique pure state extension property of an Abelian $C^*$-subalgebra of a 1-dim NCCW complex, the approximation of morphisms between two 1-dim NCCW complexes by $C^*$-diagonal preserving morphisms, and the existence of $C^*$-diagonal in inductive limits of certain 1-dim NCCW complexes.

One possible version of the Kirchberg—Phillips theorem states that simple, separable, nuclear, purely infinite C*-algebras are classified by KK-theory. In order to generalize this result to non-simple C*-algebras, Kirchberg first restricted his attention to those that absorb the Cuntz algebra O2 tensorially. C*-algebras in this class carry no KK-theoretical information in a strong sense, and they are classified by their ideal structure alone. It should be mentioned that, although this result is in Kirchberg’s work, its full proof was first published by Gabe. In joint work with Gábor Szabó, we showed a generalization of Kirchberg's O2-stable theorem that classifies G-C*-algebras up to cocycle conjugacy, where G is any second-countable, locally compact group. In our main result, we assume that actions are amenable, sufficiently outer, and absorb the trivial action on O2 up to cocycle conjugacy. In very recent work, I moreover show that the range of the classification invariant, consisting of a topological dynamical system over primitive ideals, is exhausted for any second-countable, locally compact group.

In this talk, I will recall the classification of O2-stable C*-algebras, and describe their classification invariant. Subsequently, I will give a short introduction to the C*-dynamical working framework and present the classification result for equivariant O2-stable actions. Time permitting, I will give an idea of how one can build a C*-dynamical system in the scope of our classification with a prescribed invariant.