Fourier-Stieltjes algebras of locally compact groupoids

This paper gives a first step toward extending the theory of Fourier-Stieltjes algebras from groups to groupoids. If G is a locally compact (second countable) groupoid, we show that B(G), the linear span of the Borel positive definite functions on G, is a Banach algebra when represented as an algebra of completely bounded maps on a C^*-algebra associated with G. This necessarily involves identifying equivalent elements of B(G). An example shows that the linear span of the continuous positive definite functions need not be complete. For groups, B(G) is isometric to the Banach space dual of C^*(G). For groupoids, the best analog of that fact is to be found in a representation of B(G) as a Banach space of completely bounded maps from a C^*-algebra associated with G to a C^*-algebra associated with the equivalence relation induced by G. This paper adds weight to the clues in the earlier study of Fourier-Stieltjes algebras that there is a much more general kind of duality for Banach algebras waiting to be explored.Comment: 34 page

Representations of classical Lie groups and quantized free convolution

We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition. This leads to two operations on measures which are deformations of the notions of the free convolution and the free projection. We further prove that if one replaces counting measures with others coming from the work of Perelomov and Popov on the higher order Casimir operators for classical groups, then the operations on the measures turn into the free convolution and projection themselves. We also explain the relation between our results and limit shape theorems for uniformly random lozenge tilings with and without axial symmetry.Comment: 43 pages, 4 figures. v3: relation to the Markov-Krein correspondence is updated and correcte

L\'evy Processes on Quantum Permutation Groups

We describe basic motivations behind quantum or noncommutative probability, introduce quantum L\'evy processes on compact quantum groups, and discuss several aspects of the study of the latter in the example of quantum permutation groups. The first half of this paper is a survey on quantum probability, compact quantum groups, and L\'evy processes on compact quantum groups. In the second half the theory is applied to quantum permutations groups. Explicit examples are constructed and certain classes of such L\'evy processes are classified.Comment: 60 page