Quantum symmetries in free probability

The framework of this thesis is Voiculescu's free probability theory. The main theme is the application of bialgebras, particularly Woronowicz-Kac C*-algebraic compact quantum groups, in free probability. A large part of this thesis is concerned with the class of "easy" compact quantum groups, introduced by Banica and Speicher. After a brief background section, we construct two new series Hsn, Hsn of easy quantum groups and establish some classification results. In Chapter 4, we present a unified approach to de Finetti type results for the class of easy quantum groups. In this way we recover the classical results of de Finetti and Freedman on exchangeable and rotatable sequences, and the recent free probability analogues of Kostler-Speicher and Curran for quantum exchangeable and quantum rotatable sequences, within a common framework. In Chapter 5 we introduce a notion of quantum spreadability, defined as invariance under certain objects Ai(k, n) which we call quantum increasing sequence spaces, and establish a free analogue of a famous theorem of Ryll-Nardzewski. We then consider some well-known results of Diaconis-Shahshahani on the limiting distribution of Tr(Uk), where U is uniformly chosen from On or Sn, within the context of easy quantum groups. We recover their results and establish some surprising free analogues. In Chapter 7 we consider the limiting distribution of UNANU*N and BN, where AN and BN are matrices with entries in an arbitrary C*-algebra B and UN is a quantum Haar unitary random matrix. We show that these are asymptotically free with amalgamation over B if AN and BN have limiting distributions as N goes to infinity, and that this may fail for classical Haar unitaries if B is infinite-dimensional. In the final chapter we use (non-coassociative) infinitesimal bialgebras to prove analytic subordination results in free probability.