Paradoxical decompositions and finite/infinite dichotomies in operator algebras.

Thursday, October 11, 2018 - 1:30pm
Wexler 206


Timothy Rainone
School of Mathematical and Statistical Sciences


Paradoxical decompositions appear in the work of Hausdorff, Banach, and Tarski, and perhaps go as far back as Galileo who noted that an infinite set X can be partitioned into two disjoint subsets; each having the same cardinality as X. In this talk we will explore such decompositions as they pertain to groups and group actions, and subsequently tie paradoxical phenomena (or the lack thereof) to ideas of finiteness and infiniteness in the realm of operator algebras. We will introduce the theory of bounded operators on Hilbert space and study various examples of concrete C*-algebras, emphasizing the finite/infinite nature of each example. We will end our discussion by studying the stably finite/purely infinite divide for certain C*-algebras that arise from directed graphs and dynamical systems. This talk is aimed at a general math audience.