Exotic Group C*-algebras, Tensor Products, and Related Constructions
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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.
Cite this work
Matthew Wiersma (2016). Exotic Group C*-algebras, Tensor Products, and Related Constructions. UWSpace. http://hdl.handle.net/10012/10548