Fall 2022

**September 12: George Elliott (University of Toronto)**

**Title**:*How do C*-algebras behave when you are not looking*

**Abstract**: There are two kinds of C*-algebras---moderately well-behaved ones, sometimes called amenable (the word suggests that), and those that are so far completely unfathomable. We have no idea whatsoever what they are doing, whether we are looking or not! Fortunately, in natural settings they don't appear. But even with the amenable ones you have to be careful. It's not like von Neumann algebras where completely arbitrary amenable ones (in the von Neumann algebra sense) are under total control (classified by a simple invariant and detailed structure known). (For both von Neumann algebras and C*-algebras one needs a countability restriction.) With amenable C*-algebras, there are those that are extra well behaved, a very natural and robust condition, and these also are completely understood (classified by a simple invariant, and detailed structure known---although not yet everything about group actions). By extra well behaved, I mean also simple, i.e. only trivial closed two-sided ideals. (For von Neumann algebras there is no need to restrict to this case, although it is in fact sufficient!) I also mean satisfying a certain homological condition, which it is believed always holds for amenable C*-algebras (the Universal Coefficient Theorem, or UCT)---and is in many cases known to hold. The main meaning of being extra well behaved is absorbing tensorially a very natural C*-algebra, with same invariant as the complex numbers---sometimes called the non-type I complex numbers! Since this algebra, referred to as the Jiang-Su algebra, absorbs itself, any simple amenable C*-algebra therefore becomes classifiable after tensoring with this algebra. In fact, the invariant in question, suitably defined, is unchanged after the tensoring operation, which shows it can't work in the non Jiang-Su absorbing setting, which has been observed to occur (not easy). Interestingly, not only is the invariant rather simple, but all conceivable values of it occur. This yields a second, independent, proof that it can't work beyond the class of algebras that it has been shown to classify. Of course, it is interesting to consider also non-simple algebras, for which some results are known, and also to consider amenable simple C*-algebras that are not extra well behaved, i.e., not classifiable by the invariant mentioned---a little is known in this case, but results are thin on the ground.**September 26, October 3, 10: Zhuang Niu (University of Wyoming)**

**Title**:*The classification of C*-algebras*

**Abstract**: The classification program of C*-algebras asks when a class of C*-algebras is classified by the K-theoretical invariant. I will give a brief description of the classification functor and the Elliott invariant, and then give an overview of the current classification theorem.**October 17: Vince Ruzicka (University of Wyoming)**

**Title**:*From Bratteli diagrams to dynamical systems and back*

**Abstract**: We describe the one-to-one correspondence between equivalence classes of essentially simple ordered Bratteli diagrams and isomorphism classes of essentially minimal dynamical systems, when our topological space is compact, metrizable and totally disconnected. We also briefly discuss some motivation for finding such an equivalence. The talk is based on a work by Herman, Putnam, and Skau in the early 90's.**October 24: Andrew Toms (Purdue University)**

**Title**:*Homotopy groups of Cuntz classes*

**Abstract**: We consider the sets of positive elements with fixed Cuntz class in certain C*-algebras of real rank zero. These algebras include the irrational non-commutative tori and AF algebras. We show that when the Cuntz class is not compact, its homotopy groups vanish. Combined with work of Zhang for the compact case, this gives a complete calculation of the homotopy groups for these classes.**October 31: Zach Pence (University of Wyoming)**

**Title**:*Simplicity of C*-algebras associated to minimal dynamical systems*

**Abstract**: In this talk we survey some basic properties of minimal dynamical systems and an associated C*-algebra. In particular we look at a theorem claiming the simplicity of these algebras. This is due to the work of S.C. Power in the 1970's.**November 7: Ping Zhong (University of Wyoming)**

**Title**:*Brown measure of a sum of two free random variables and deformed random matrix models*

**Abstract**: Random variables in Voiculescuâ€™s free probability theory can model the limits of suitable random matrix models. The Brown measure of a random variable in free probability is a replacement for the eigenvalue counting measure of square matrices. The R-diagonal operators are a large family of free random variables that are natural limit operators of various well-known non-normal random matrix models. I will report some recent progresses on the Brown measure of the sum of two free random variables, one of which is circular, elliptic, or R-diagonal. We show that subordination functions in free additive convolution can detect information about the Brown measure. These Brown measures are related to the limit eigenvalue distributions of deformed i.i.d., GUE, elliptic, and single ring random matrix models. The talk is an introduction to my recent work on Brown measures available at arXiv:2108.09844, arXiv:2209.11823 (joint with Belinschi and Yin), and arXiv:2209.12379 (joint with Bercovici)**November 14: Tyrrell McAllister (University of Wyoming)**

**Title**:*Characterizations of ellipsoidal cones in normed vector spaces*

**Abstract**: Partially-ordered linear spaces are important objects of study in fields ranging from economics (as commodity spaces) to special relativity (as Minkowski spacetime). In each of these cases, understanding the space depends upon understanding a particular geometric object within it, namely the convex cone that defines the partial order. We give two characterizations of those convex cones that are ellipsoidal cones. The first states that a cone is ellipsoidal if and only if each of its bounded sections is centrally symmetric. The second states that a cone $C$ is ellipsoidal if and only if the intersection of the boundary of $C$ with the boundary of $x - C$ is contained in a hyperplane for all $x$ in the interior of $C$.

Spring 2023

**February 13, 20 : Vince Ruzicka (University of Wyoming)**

**Title**:*Some convenient properties of C*-algebras*

**Abstract**: In this talk, we describe some classical results about C*-algebras. We start by giving some details of the proofs for the existence of the continuous function calculus and the spectral mapping theorem, for self-adjoint elements. We then describe the order structure of a C*-algebra and its dual before concluding with a brief discussion of the classification of abelian algebras.**February 27, March 6: Zach Pence (University of Wyoming)**

**Title**:*Z-stable C*-algebras*

**Abstract**: In this talk we look at the Jiang-Su algebra Z and Z-stable C*-algebras. The Jiang-Su algebra was discovered in the context of the classification program, so we begin by motivating it in this context, in particular why Z-stability is required for a C*-algebra to be classifiable by the Elliott invariant. With this motivation in hand, we will proceed to define Z and explore some of its properties.**April 3: Vince Ruzicka (University of Wyoming)**

**Title**:*Progress on the generator problem for C*-algebras*

**Abstract**: The generator problem for C*-algebras asks about the least number of elements required to generate a given algebra. Von Neumann initiated the study of this problem by showing that abelian von Neumann algebras are singly generated, and later on the problem made it onto Kadison's famous list. In this talk, we will describe the history of the generator problem along with some results due to Hannes Thiel published in the last few years---in particular, that all classifiable C*-algebras are generically generated. We conclude by discussing the generators of algebras that don't fall within the current classification program.**April 20 (Fisk Lecture): Gene Abrams (University of Colorado, Colorado Springs)**

**Title**:*Morita equivalence for graded rings*

**Abstract**: The classical Morita Theorem for rings established the equivalence of three statements, involving categorical equivalences, isomorphisms between corners of finite matrix rings, and bimodule homomorphisms. A fourth equivalent statement (established later) involves an isomorphism between infinite matrix rings. I'll spend the first part of this talk describing the ideas involved, and some of the history of the classical Morita Theorem. I'll then describe our two main results, in which we establish the equivalence of analogous statements involving two types of graded categorical equivalences, graded isomorphisms between corners of finite matrix rings, graded bimodule homomorphisms, and graded isomorphisms between infinite matrix rings. I'll also describe some connections between these results and results about C*-algebras. Only a basic level of ring theory background will be assumed. This is a joint work with Efren Ruiz and Mark Tomforde.**April: Marco Merkli (Memorial University of Newfoundland)**

The seminar in general is on Mondays, 16:10--17:00 at RH 247. If you like to contribute a talk, please contact me.