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Abstract: We consider nonlocal games (these will be introduced), but allow for quantum questions and answers living in quantum sets. It turns out the typical definitions from the nonlocal games literature can be extended and many of the typical results still hold in this quantum setting. The analogies to the typical setting can be seen clearly using string diagrams!

This talk contains many diagrams. Never fear: We begin by motivating the diagrams using multimatrix algebras. We use string diagrams to introduce extended definitions of games, correlations, and synchronicity. We show that perfect correlations and strategies for synchronous quantum games must be synchronous. Time permitting, as a key example: When the graph homomorphism game is extended to quantum graphs, we get a synchronous game, whose perfect correlations correspond to quantum graph homomorphisms! The talk is based on a preprint Quantum games and synchronicity. This work is inspired by Musto, Reutter, and Verdon's paper A compositional approach to quantum functions, and relies heavily on the reference Categories for Quantum Theory by Heunen and Vicary for string diagrams in quantum information.

Abstract: We begin by characterizing the KMS states of the Toeplitz C*-algebra of the ax+b monoid in terms of a class of probability measures on the unit circle satisfying a 'subconformal' property. We then give explicit formulas for the atomic subconformal measures and prove a 'multiplicative' version of Wiener's lemma that implies that Lebesgue measure is the only nonatomic one.

Abstract: I will begin by describing the Toeplitz C*-algebra of the ax+b monoid over the natural numbers introduced in recent joint work with an Huef and Raeburn, and then compute the equilibrium states for inverse temperature below 1 by first transforming the problem into the computation of probability measures on the unit circle that satisfy a certain subconformal property. This is joint work with Tyler Schulz.

Abstract: Let $G$ be a countable discrete group, and let $\mu$ be a probability measure on $G$ with finite (Shannon) entropy. We initiate the study of several related concepts associate to a probability measure $\mu$ and exploit their relations. First, we look at the concept of {\it Lyapunov exponent} of $\mu$ with respect to weights on $G$ and build a framework that connects it to the entropy of $\mu$ in $G$. This is done by introducing a generalization of Avez entropy $h(G,\mu)$, taking into account the given weight, and investigating in details their relations together as well as to the actions of $G$ on measurable stationary spaces.

As a byproduct of our techniques, we show that for a large class of groups (e.g. groups with rapid decay) and probability measures on them, the weak containment of the representation $\pi_X$ of $G$ on a $\mu$-stationary space $(X,\xi)$ implies that

$$h(G,\mu)=h_\mu(X,\xi),$$

where $h_\mu(X,\xi)$ is the Furstenberg entropy of $(X,\xi)$.

This implies that these types of stationary spaces are precisely measure-preserving extension of the Poisson boundary of $(G,\mu)$. In particular, $(X,\xi)$ is an amenable $(G,\mu)$-space if and only if it is a measure-preserving extension of the Poisson boundary of $(G,\mu)$.

This is a join work with Benjamin Anderson-Sackaney, Tim de Laat, and Matthew Wiersma.

(joint work with C. Eagle, I. Goldbring, and R. Miller)

Abstract: I will discuss recent work on the effective (computable) content of K-theory for operator algebras. I will explain at least the architecture of the proofs of the following theorems. 1) If $A$ is a computably presentable unital $C^*$-algebra, then the Abelian group $K_0(A)$ has a c.e. (a.k.a recursive) presentation. 2) If $A$ is a computably presentable uniformly hyperfinite algebra, then $K_0(A)$ has a computable presentation; moreover, the trace of $A$ is computable and its supernatural number is lower semi-computable. 3) UHF algebras are computably categorical; that is any two of their computable presentations are computably *-isomorphic.

Abstract: Topological full groups arise from dynamical systems. Their main interest is for group theorists. But they first appeared from considerations in operator algebras. I'll will explain these topics. No background is needed.