Perturbations of Operator Algebras and Related Topics

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This project has two main areas of focus. The first of these concerns the perturbation theory of operator algebras with particular reference to the longstanding Kadison-Kastler problem, which asks whether operator algebras that are close in a suitable metric must be isomorphic (or even unitarily equivalent). This is part of a more general and very difficult problem: when two algebras are the same (isomorphic), how can we recognize that this is so? The difficulty arises because the same algebra can have many different, seemingly unrelated, representations. In recent joint work with coauthors, the principal investigator has made considerable progress on the Kadison-Kastler problem, and he plans to continue this work in several directions: an isomorphism result for close crossed product algebras, the K-theory of close algebras, the shared properties of close algebras, and the question of whether close containments lead to embeddings. Recent progress has come for algebras that have the similarity property, and this has a reformulation in cohomological terms. Thus the second main area of focus will be the Kadison-Ringrose problem, which asks whether certain cohomology groups (which act as obstructions to desirable properties) must vanish. This will involve the theory of completely bounded linear and multilinear maps and also connects to the bounded projection property.The modern study of operator algebras has evolved from two main sources. Matrices, which are generalizations of numbers, were introduced to solve equations and now find applications from computer graphics to search engines for the internet. In formulating quantum mechanics mathematically, von Neumann found that he needed infinite-dimensional versions of matrices called linear operators, which were best studied in operator algebras. Moreover, the time-evolution of quantum mechanical systems came to be expressed in terms of the crossed product by groups of automorphisms. The emerging theory of quantum computation is substantially based on the theory of completely bounded and completely positive maps at the matrix level, and these topics underlie much of the work that will be undertaken. Thus, the results obtained in these areas are likely to impact some of these more concrete areas, since the finite factors are those operator algebras that most closely model matrix algebras. An important aspect of the principal investigator's work has been the training of postdoctoral researchers and doctoral students. Most of the young people who have been mentored by the principal investigator are now in faculty positions where they are training the next generation of scientists and engineers. A scientifically and mathematically trained workforce is essential for the technological future of the country, so the principal investigator will continue to give a central role to the mentoring of young mathematicians.