Activities This Week

I will tell you about a nice little paper of Pisier describing a rather general approach ubiquitous in my corner of mathematics. Namely, “if you want to know more about an object consider matrices over it”. The applications of this result are to strong convergence, such as results of Haagerup and Thorbjornsen and Collins and Male.

Link to paper

A function f on a countable group G forms a cocycle for the left regular representation of G if for every g in G, the difference between f and its left g-translate belongs to ℓ²(G). I will show that although the left and right regular representations are unitarily equivalent in general, their corresponding spaces of cocycles can be genuinely different. Using special asymmetric cocycles, I will explain how to construct Bernoulli schemes over amenable groups whose left dynamics differ substantially from their right dynamics. This is joint work with Zemer Kosloff.

Laurent Schwartz showed that a hermitian kernel K is the difference of two positive kernels iff it has an associated reproducing kernel Krein space of functions. We generalize this equivalence to the free non-commutative setting of nc functions\kernels, with possible application to the locally bounded (equivalently, uniformly nc analytic) case (generalizing a result of Alpay in the classical commutative setting).