Mathematical Sciences: The K-Theory and Isomorphism Invariants of Toeplitz Algebras

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The situation of a group acting on a space, for instance the group of real numbers acting by time evolution on the state space of a physical system, is encountered quite often in mathematics and its applications. In the physics example, every possible configuration of the system lies on a line flowing from remote past to distant future. One is interested in what happens overall, in the long run, which may be quite difficult to infer solely from what happens over short intervals of time. For purposes of analysis, the available information can be encoded in several different ways: by differential equations or vector fields or foliations or, as is the case in Professor Xia's research project, by an algebra of operators on Hilbert space. Professor Xia will study a rather general Toeplitz algebra construction that has recently been developed for (non- commutative) dynamical systems; this is related to the more familiar notion of a crossed product. The specific objectives of his work are to calculate the K-groups of Toeplitz algebras and their associated commutator ideals, and to see whether the Toeplitz algebra is a complete isomorphism invariant of the underlying dynamical system.