Noncommutative Geometry Seminar

Department of Mathematics, University of Hawaii
2565 McCarthy Mall, Keller 401A
Honolulu HI 96822

Cartan subalgebras in C*-algebras

We will follow the article of Jean Renault, Cartan subalgebras in C^*-algebras, and try to present the following result: For every pair (A,B) where A is a C^*-algebra and B a Cartan subalgebra of A is of the form ( C_r^* (G,A) , C_0( G^{(0)} ) ) , where G is an étale topologically principal Hausdorff groupoid over G^{(0)} , and A a twist over G.

Cartan subalgebras in C*-algebras II

We will present some results on pseudogroups of partial homeomorphisms acting on topological spaces, and their relationship to étale groupoids. This will allow us to construct out of any Cartan pair (A,B), an étale groupoid such that its C*-algebra is *-isomorphic to A.

Comments on the coarse geometry of ‘box’ spaces

Box spaces built from residually finite discrete groups are a rich source of examples and counterexamples in coarse geometry and operator algebra K-theory. In two talks, I will explain some aspects of the coarse geometry of box spaces, and the relation to analytic, geometric and algebraic properties of discrete groups.

Amenable traces

A state t on a C*-algebra A is a trace if t(ab)=t(ba) for all a,b in A. A trace is called amenable if it can be approximated by the usual traces on matrices in some sense (and is thus relatively tractable). I’ll discuss the connection to invariant means of groups, and some other invariant mean type examples, time permitting.

Amenable traces on uniform Roe algebras

Uniform Roe algebras are C*-algebras that capture the large-scale geometry of metric spaces. I’ll discuss why such traces always come from invariant means (in an appropriate sense), and why they are always amenable in the sense discussed last time. Time permitting, I’ll also discuss some ‘pathological' examples.

Introduction to Kazhdan's Property (T)

We will introduce Property (T) for finitely generated groups. This is an approximation property which is a strong negation of amenability. We will give examples of groups with property (T), and prove Guichardet's theorem: in this case, G has (T) iff every isometric affine action of G on a Hilbert space has a fixed point.

SL(3,Z) has Kazhdan's Property (T)

We will proved the result above, providing an example of infinite group with property T.

Property (T), Laplacians, and Kazhdan projections

I'll continue where Clément left off and say some more about consequences of property (T) for group C*-algebras, partly in terms of Laplace operators and so-called Kazhdan projections. Time permitting, I'll also sketch how one can give a computer-assisted proof that SL(3,Z) has property (T) that is quite different from the one Clément presented.

Gromov's theorem for groups of polynomial growth

Speaker: Sasha Markovichenko- Part one of the talk will introduce the elementary concepts of growth, some history as well as a proof of Gromov's theorem " Polynomial growth implies virtually nilpotent" assuming the hypothesis of Milnor's lemma is satisfied for groups of polynomial growth.

Speaker: Umar Gaffar- Part two of the talk will introduce the elementary concepts of nonstandard analysis including the nonstandard hull construction. We will also show how the nonstandard hull is used to guarantee that a group of polynomial growth satisfies the hypothesis of Milnor's lemma, thus completing the proof of Gromov's.

Classification and the UCT

The goal of the talk is to explain what is the UCT, and why it is so important in classification of C*-algebras. No background in KK-theory or classification is assumed. To motivate the classification program, we will start by Elliot's classification of AF-algebras. Then we will define Elliot's invariant and state the most general (to my knowledge) classification result: separable unital simple UCT C*-algebras finite nuclear dimension are classified by Elliot's invariant. Then we'll discussed strategies to remove the UCT hypothesis.