We consider positive operator-valued measures whose image is the bounded operators acting on an infinite-dimensional Hilbert space, and we relax, when possible, the usual assumption of positivity of the operator-valued measure seen in the quantum information theory literature. We consider integrals of quantum random variables, extending work done by Farenick--Plosker--Smith (2011) to the setting of infinite-dimensional Hilbert space, and develop positive operator-valued versions of a number of classical measure decomposition theorems. This is joint work with D. Mclaren and C. Ramsey.

I will explain further details about the talk one week before, in particular on Tingley's problem for operator algebras.

Operator systems (self-adjoint unital subspaces of C*-algebras) had been looked as early as the 1970's. As early as the 1980's, operator algebraists realized that there was a rich theory in noncommutative convexity (matrix convexity). Though it was not until 1999 that it was noticed by Webster and Winkler that operator systems and compact matrix convex sets were intimately connected. Using Webster-Winkler duality and noncommutative Choquet theory, we have been able to present a new way in looking at Choquet points of finite-dimensional compact matrix convex sets. We will begin by reviewing noncommutative convex theory and Webster-Winkler duality. Operator system tensor products will be reviewed (if needed). As time permits we will then discuss Choquet points of finite-dimensional compact matrix convex sets. This is joint work with Adam Dor-On and Thomas Sinclair.

In the 1940's Krein asked the question whether Lipschitz functions f: R -> C are also operator Lipschitz functions in the sense that the mapping B(H)_sa -> B(H): x -> f(x) is Lipschitz. The answer to this question is negative unless additional smoothness assumptions are imposed on f. On the other hand if the uniform norm on B(H) is replaced by the Schatten Lp-norm then every Lipschitz function is operator Lipschitz (Potapov-Sukochev 2010). In this talk we give a sharp proof of this result through so-called end-point estimates (weak L1-estimates and BMO-estimates). In order to achieve this we further develop the theory of Markov dilations and De Leeuw theorems. This is joint work with D. Potapov, F. Sukochev and D. Zanin.

A ``non-local game'' consists of two players, who are each provided questions from a referee and then supply answers. The game comes with rules which determine if the answers supplied by the players are correct or not. The players cooperate to win each round of the game, but the ``non-locality'' of the games means that the players cannot communicate by classical means during each round of the game. They can, however, agree upon a shared strategy for producing satisfactory answers. Non-local games are of interest in quantum information theory because quite often the only winning strategy is a so-called quantum strategy - i.e., one which utilizes some shared quantum entanglement between the players as a resource. In this talk, I will focus on a particular class of non-local games, called synchronous games. For these games one can associate a certain associative algebra A whose structure completely characterizes the existence of winning deterministic and probabilistic (quantum) strategies for these games. As a particular example, I will focus on the graph isomorphism game, which takes as inputs two graphs, and is devised so that a winning deterministic strategy requires that the two graphs be isomorphic. On the other hand, a probabilistic winning strategy relaxes this condition to the two graphs being what quantum information theorists call ``quantum isomorphic''. I will explain how the notion of quantum isomorphism mentioned above is intimately connected to the theory of Hopf bi-Galois objects: Two graphs are quantum isomorphic if and only if the game algebra A is a Hopf bi-Galois extension over the universal Hopf algebras coacting on the function algebras of the two graphs. I will explain how this Hopf-algebraic interpretation of the graph isomorphism game provides some fundamental new insights. **NOTE**: There will be a sequel to this talk given by Kari Eifler (TAMU) at 4pm in the Linear Analysis Seminar. Both talks will be complementary, yet self-contained.
Abstract