UC Berkeley

The framework of this thesis is Voiculescu's free probability theory. The main theme is the application of bialgebras, particularly Woronowicz-Kac 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 "hyperoctahedral" series 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 Koestler-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 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 the trace of the k-th power of U, where U is uniformly chosen from O(N) or S(N), within the context of easy quantum groups. We recover their results and establish some surprising free analogues. In Chapter 7 we prove asymptotic freeness results for Haar quantum unitary random matrices. In the final chapter we use (non-coassociative) infinitesimal bialgebras to prove analytic subordination results in free probability.