Support Expansion Operator Algebras

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.