Sam Harris, Pure Mathematics, University of Waterloo
"Unitary Correlation Sets"
In this talk we consider the images of states on generators of certain tensor products of $U_{nc}(n)$, where $U_{nc}(n)$ is the universal $C^{\ast}$-algebra generated by entries of an $n \times n$ unitary matrix. These sets associated with tensor products of $U_{nc}(n)$ are called unitary correlation sets. We will discuss different models for these correlations, including the spatial model and the commuting model. Akin to Tsirelson's problem, we will show that Connes' embedding problem is equivalent to deciding whether every correlation in the commuting model can be approximated by correlations in the spatial model. Moreover, these sets define cross norms on $M_n \otimes M_n$, and hence Connes' embedding problem is equivalent to deciding whether or not these cross norms on $M_n \otimes M_n$ are equal for all $n \geq 2$.
MC 5417