Approximation Theory and Operator Algebras

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BrownThe investigator will continue studying approximation properties of operator algebras, concentrating on fundamental questions arising in C*- and W*-algebra theory. For example, Elliott's classification program took a dramatic turn when counterexamples were constructed by Rordam and Toms. However, recent work of Winter strongly suggests that the classification of algebras which are "finite dimensional" (in a suitable noncommutative topological sense) should be classifiable. We will investigate the classification problem for these algebras, but utilizing a different invariant -- one that turns out to be functorially equivalent to the classical Elliott invariant, but formally carries much more information. In a W*-direction, we will continue our study of topological spaces associated to II_1 factors which admit microstates (in the sense of Voiculescu). These topological invariants haven't yet been systematically investigated, so it's quite natural to explore them. One very successful idea in mathematics is that we can learn about complicated objects by approximating with simpler objects, then passing to a limit. For example, in calculus we compute the area under a curve using rectangular approximations, then refining the approximations over and over. Operator algebras are (usually) infinite dimensional objects which provide the natural framework for quantum mechanics, for example. Moreover, deep and unexpected connections with other areas of mathematics such as geometry, topology and probability were discovered over the years. As such, a solid understanding of the structure of operator algebras is important. The general philosophy of using approximations by simpler objects becomes especially relevant here since the objects of interest are infinite dimensional. The investigator will continue an established tradition of trying to use finite dimensional approximations to better understand some fundamental infinite dimensional objects.