(Quantum) discreteness, spectrum compactness and uniform continuity

We prove a number of results linking properties of actions by compact groups (both quantum and classical) on Banach spaces, such as uniform continuity, spectrum finiteness and extensibility of the actions across several constructions. Examples include: (a) a unitary representation of a compact quantum group induces a continuous action on the $C^*$-algebra of bounded operators if and only if it has finitely many isotypic components, and hence is uniformly continuous; (b) a compact quantum group is finite if and only if its continuous actions on $C^*$-algebras lift to continuous actions on either the multiplier algebras or von Neumann envelopes thereof; (c) a (classical) locally compact group $\mathbb{G}$ is discrete if and only if the forgetful functor from $\mathbb{G}$-acted-upon compact $T_2$ spaces back to compact $T_2$ spaces creates coproducts; (d) a representation of a linearly reductive quantum group has finitely many isotypic components if and only if its restrictions to two topologically-generating quantum subgroups, one of which is normal, do; (e) equivalent characterizations of uniform continuity for actions of compact groups on Banach spaces, e.g. that such an action is uniformly continuous if and only if its restrictions to a pro-torus and to pro-$p$ subgroups are.Comment: 20 pages + reference

The asymptotic shape theorem for generalized first passage percolation

We generalize the asymptotic shape theorem in first passage percolation on $\mathbb{Z}^d$ to cover the case of general semimetrics. We prove a structure theorem for equivariant semimetrics on topological groups and an extended version of the maximal inequality for $\mathbb{Z}^d$-cocycles of Boivin and Derriennic in the vector-valued case. This inequality will imply a very general form of Kingman's subadditive ergodic theorem. For certain classes of generalized first passage percolation, we prove further structure theorems and provide rates of convergence for the asymptotic shape theorem. We also establish a general form of the multiplicative ergodic theorem of Karlsson and Ledrappier for cocycles with values in separable Banach spaces with the Radon--Nikodym property.Comment: Published in at http://dx.doi.org/10.1214/09-AOP491 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Automorphism groups of randomized structures

We study automorphism groups of randomizations of separable structures, with focus on the $\aleph_0$-categorical case. We give a description of the automorphism group of the Borel randomization in terms of the group of the original structure. In the $\aleph_0$-categorical context, this provides a new source of Roelcke precompact Polish groups, and we describe the associated Roelcke compactifications. This allows us also to recover and generalize preservation results of stable and NIP formulas previously established in the literature, via a Banach-theoretic translation. Finally, we study and classify the separable models of the theory of beautiful pairs of randomizations, showing in particular that this theory is never $\aleph_0$-categorical (except in basic cases).Comment: 28 page

On the bounded cohomology of semi-simple groups, S-arithmetic groups and products

We prove vanishing results for Lie groups and algebraic groups (over any local field) in bounded cohomology. The main result is a vanishing below twice the rank for semi-simple groups. Related rigidity results are established for S-arithmetic groups and groups over global fields. We also establish vanishing and cohomological rigidity results for products of general locally compact groups and their lattices