Support Expansion Operator Algebras
Eisner, Joseph, Mathematics - Graduate School of Arts and Sciences, University of Virginia
The collection of bounded operators which have at most finitely many nonzero entries in each row and column of their standard array forms a *-subalgebra of B(ell^2) and thus their norm closure a C*-algebra. We generalize this construction in several directions and settings, giving rise to a very general procedure for constructing concrete support expansion C*-algebras over a represented tracial von Neumann algebra. We go on to give a thorough analysis of the containment poset of concrete support expansion C*-algebras when the von Neumann algebra is taken to be \ell^\infty inside of B(\ell^2) and when it is taken to be L^\infty(R) in B(L^2(R)). In particular we will show the containment poset of support expansion C*-algebras over L^\infty(R) in B(L^2(R)) has uncountable ascending and descending chains as well as uncountable antichains.
The C*-algebra discussed in our first sentence is naturally realized as a uniform Roe algebra. In the second half of this dissertation we use measurable and quantum relations per "Quantum Relations" (Weaver 2012) along with our home-baked intermediary cantankerous relations to define measurable, cantankerous and quantum uniform Roe algebras. We then realize the support expansion C*-algebras we developed in the first half as uniform Roe algebras in an appropriate sense.
PHD (Doctor of Philosophy)
operator algebra, Roe algebra, functional analysis
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