Activities This Week

The noncommutative (nc) disc algebra $\mathcal{A}_d$ was studied extensively first by Popescu. It is the norm closed operator algebra generated by the left creation operators on the full Fock space. This algebra is semi-Dirichlet. Namely, $\mathcal{A}_d^* \mathcal{A}_d \subset \overline{\mathcal{A}_d + \mathcal{A}_d^*} = \mathcal{S}_d$. Therefore, one can perform a GNS type construction to obtain representations of $\mathcal{A}_d$ from states on $\mathcal{S}_d.$ This observation is one of the ingredients in the nc Clark theory developed by Jury and Martin.

In this talk, I will focus on nc rational functions and, in particular, inner ones. I will show how one obtains from an nc inner rational a finitely-correlated state (Bratteli and Jorgensen) on the Cuntz algebra. Connect the finitely-correlated states to minimal isometric dilations of finite-dimensional row coisometries and the work of Davidson, Kribbs, and Shpigel. Lastly, I will show that many finitely-correlated states are peak states in the sense of Clouatre and Thomson.

This talk is based on joint works with Mike Jury and Rob Martin.

(A joint work with Waltraud Lederle) In order to avoid technicalities I will focus on one specific example for a higher $\mathbb{Q}$-rank lattice: the group $\Gamma = \mathrm{SL}_3(\mathbb{Z})$. This group exhibits strong rigidity properties, some of which are naturally expressed in topological terms. For example, one of the earliest rigidity results, the congruence subgroup property which was established independently by Mennicke and Bass-Milnor-Serre, can be expressed as an equality between two group topologies on $\Gamma$: The profinite and the congruence topologies. Margulis’ celebrated normal subgroup theorem can be thought of as the statement that even the normal topology coincides with these two. Here the normal topology is defined by taking all infinite normal subgroups as a basis of identity neighborhoods for a topology on $\Gamma$. Together with Waltraud Lederle we introduce an a-priori much finer topology on $\Gamma$ called the boomerang topology and show that in fact even this topology coincides with the congruence topology. As a result we obtain a generalization of a rigidity theorem for probability measure preserving actions due to Nevo-Stuck-Zimmer.