New directions in C*-algebras, Groupoids, and Inverse Semigroups

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Von Neumann introduced operator algebras to express quantum mechanics in a mathematical language. C*-algebras are a tractable and rich class of operator algebras, and their study has a geometric flavour---they have been described by Connes as "noncommutative geometry". Geometry is governed by its beautiful symmetries, and my program explores the extent to which C*-algebras are governed by "partial symmetries" through structures called groupoids and inverse semigroups. A groupoid is certain type of set of partial symmetries on some geometric space. To every groupoid, one may construct a C*-algebra built from it. My long-term goal is to determine whether every C*-algebra is obtainable from partial symmetries in this way; or if this is not possible, describe the largest class of C*-algebras obtainable in this way. If all or most are obtainable in this way, it would foster new research connections between the theories of operator algebras, dynamical systems, and semigroups. In 1976, the Canadian mathematician George Elliott made a major discovery: that some classes of C*-algebras are completely determined by their K-theory group. The K-theory group of a C*-algebra is a kind of listing of its (possibly infinite-dimensional) subspaces arranged in order according to their (generalized) dimension. Expanding the classes for which Elliott's result hold has been an ongoing program ever since---the "Elliott Classification Program." Giving efficient methods for computing the K-theory groups of groupoid C*-algebras would be of great value to this program, especially if work on our long term goal shows that most C*-algebras are obtained this way. It is then a short-term goal to develop such methods, and apply them to examples of recent interest, including those arising from chaotic dynamical systems and finite automata.