Mathematical Sciences: Operator Algebras and Their Applications

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Professors Effros, Popa, and Takesaki will continue their investigations into operator algebra theory and its applications. Effros will focus on matricial norms and orderings. This is expected to lead to new approaches to Fourier analysis on non- abelian groups and to the structure theory of non-injective von Neumann algebras. Popa will work on the structure of factors, index theory for subfactors, amenability, and rigidity. Takesaki will study group actions on injective factors, and explore the relationship between modular theory for von Neumann algebras and Connes' cyclic cohomology. The Hilbert space operators that lie at the heart of this project may be thought of as a species of enriched numbers. They obey the same laws of arithmetic as numbers with two notable exceptions, namely that the result of multiplication depends on the order in which the factors are taken, and not every nonzero operator has an inverse. Mathematical analysis knows only two number systems, the real and complex, but there is an immense variety of operator algebras. Situations unsuitable for numerical representation can often be represented in some fashion by operator algebras, thanks to the greater degree of flexibility available with the latter. The price for this is that understanding the structure and classification of operator algebras requires considerable effort, as for instance the work to be undertaken in this project.