Exotic Group C*-algebras, Tensor Products, and Related Constructions

Abstract

Recently there has been a rejuvenated interest in exotic group C*-algebras, i.e., group C*-algebras which are "intermediate" to the full and reduced group C*-algebras. This resurgence began with the introduction of the class of group $L^p$-representations and their associated C*-algebras (a class of potentially exotic group C*-algebras) by Brown and Guentner. Unlike previous examples of exotic group C*-algebras, this class of examples is universally defined for all locally compact groups. In this thesis we compare this new class of exotic group C*-algebras to previously known examples of exotic group C*-algebras in several key examples and produces new examples of exotic group C*-algebras.

Similar to the definition of exotic group C*-algebras, an exotic C*-tensor product is a C*-tensor product which is intermediate to the minimal and maximal C*-tensor products. Borrowing from the theory of $L^p$-representations, we construct many exotic C*-tensor products for group C*-algebras.

We will also study the $L^p$-Fourier and Fourier-Stieltjes algebras of a locally compact group. These are ideals which of the Fourier-Stieltjes algebras containing the Fourier algebras and correspond to the class of $L^p$-representations. We study the structural properties of these algebras and classify the Fourier-Stieltjes spaces of SL(2,R) which are ideals in the Fourier-Stieltjes algebra.

There are many different tensor products considered in the category of C*-algebras. In contrast, virtually the only tensor product ever considered for von Neumann algebras is the normal spatial tensor product. We propose a definition for what a generic tensor product in the category of von Neumann algebras should be and study properties of von Neumann algebras in relation to these tensor products.