Noncommutative Analysis

Category: Seminar

Seminar talk by Dor-On: Quantum symmetries in the representation theory of operator algebras

NOTE: THE SEMINAR WAS POSTPONED TO DECEMBER 10.

On next Thursday the Operator Algebras and Operator Seminar will convene for a talk by Adam Dor-On.

Title: Quantum symmetries in the representation theory of operator algebras

Speaker: Adam Dor-On (University of Illinois, Urbana-Champaign)

Time: AFTERNOON Thursday Dec. 10, 2020 (NOTE: THE SEMINAR WAS POSTPONED BY ONE WEEK FROM ORIGINAL DATE).

(Zoom room will open about ten minutes earlier, and the talk will begin at 15:30)

Zoom link: email me.

Abstract:

We introduce a non-self-adjoint generalization of Quigg’s notion of coaction of a discrete group G on a C*-algebra. We call these coactions “quantum symmetries” because from the point of view of quantum groups, coactions on C*-algebras are just actions of a quantum dual group of G on the C*-algebra. We introduce and develop a compatible C*-envelope, which is the smallest C*-coaction system which contains a given operator algebra coaction system, and we call it the cosystem C*-envelope.

It turns out that the new point of view of quantum symmetries of non-self-adjoint algebras is useful for resolving problems in both C*-algebra theory and non-self-adjoint operator algebra theory. We use quantum symmetries to resolve some problems left open in work of Clouatre and Ramsey on finite dimensional approximations of representations, as well as a problem of Carlsen, Larsen, Sims and Vitadello on the existence of a co-universal C*-algebra for product systems over arbitrary right LCM semigroup embedded in groups. This latter problem was resolved for abelian lattice ordered semigroups by the speaker and Katsoulis, and we extend this to arbitrary right LCM semigroups. Consequently, we are also able to extend the Hao-Ng isomorphism theorems of the speaker with Katsoulis from abelian lattice ordered semigroups to arbitrary right LCM semigroups.

*This talk is based on two papers. One with Clouatre, and another with Kakariadis, Katsoulis, Laca and X. Li.

Seminar talk by Salomon: Combinatorial and operator algebraic aspects of proximal actions

The Operator Algebras and Operator Theory Seminar is back (sort of). This semester we will have the seminar on Thursdays 15:30 (Israel time) about once in a while. Please send me an email if you want to join the mailing list and get the link for the zoom meetings. Here are the details for our first talk :

Title: Combinatorial and operator algebraic aspects of proximal actions

Speaker: Guy Salomon (Weizmann Institute)

Time: 15:30-16:30,Thursday Nov. 12, 2020

(Zoom room will open about ten minutes earlier, and the talk will begin at 15:30)

Zoom link: email me.

Abstract:

An action of a discrete group G on a compact Hausdorff space X is called proximal if for every two points x and y of X there is a net g_i \in G such that \lim(g_i x)=\lim(g_i y), and strongly proximal if the natural action of G on the space P(X) of probability measures on X is proximal. The group G is called strongly amenable if all of its proximal actions have a fixed point and amenable if all of its strongly proximal actions have a fixed point.

In this talk I will present relations between some fundamental operator theoretic concepts to proximal and strongly proximal actions, and hence to amenable and strongly amenable groups. In particular, I will focus on the C*-algebra of continuous functions over the universal minimal proximal G-flow and characterize it in the category of G-operator-systems.

I will then show that nontrivial proximal actions of G can arise from partitions of G into a certain kind of “large” subsets. If time allows, I will also present some relations to the Poisson boundaries of G. The talk is based on a joint work with Matthew Kennedy and Sven Raum.