Professors Arveson and Voiculescu will investigate several areas of noncommutative analysis involving von Neumann algebras, quantized index theory, C* - algebras, Fredholm modules, and noncommutative random variables. Arveson's part of the project involves the index theory and classification theory of semigroups of endomorphisms of factors, and the structure of the C* - algebras associated to such semigroups. Voiculescu will continue his research on quasicentral approximate units relative to normed ideals and their relationship to Fredholm modules, and on the noncommutative probability theory of free products. He will look further into the relation between nuclear C* - algebras and the approximation theory of operators. This mathematical research project is concerned with various constructions and techniques involving operators on Hilbert space. Operators may be thought of as a species of enriched numbers. They obey the same rules of arithmetic as numbers except that the result of multiplication depends upon the order in which the factors are taken, and not every nonzero operator has an inverse. It has proven to be a very fruitful exercise, both in theoretical physics and in pure mathematics, to see what happens when numbers are replaced by operators in a given context. Professor Arveson, for instance, is studying systems that arise when time-dependent numbers are replaced by time-dependent operators. One of Professor Voiculescu's ongoing investigations involves a noncommutative version of probability theory in which random variables are treated as if they were operator-valued rather than number-valued.
