Proposal: DMS-9970693 Principal Investigator: Donald W. Hadwin
Abstract: This project includes the study of a generalized notion of composition operators based on unital endomorphisms of unital algebras of operators having a cyclic separating vector. This general notion unifies the analytic and measure-theoretic approaches, and it leads naturally to the study of a large collection of interesting new classes of operators. The project also includes attacks on Kadison's similarity problem from several different angles: one from a purely algebraic approach based on Pisier's new ''factorization'' results; another using invariant operator ranges involving a situation where the definition of the homomorphism is fixed but where the C*-algebras vary; and a third in terms of the hyperreflexivity problem for von Neumann algebras. Another part of this project includes a study of invariant subspaces. Here existence results will be based on new ideas of Haagerup, and stability results will extend previous joint work with John B. Conway. The project also includes a study of certain reduced free products of matrix algebras generating type III factors. Finally, the project includes further work on reflexivity and hyperreflexivity and approximate operator theory.
The theory of operators on Hilbert space is a part of mathematics that generalizes to infinite dimensions the basic ideas of matrix theory and elementary linear algebra. The growth of this subject was stimulated by and has applications to the study of integral and differential equations, quantum mechanics, and control theory. More recently, operator theory has interacted directly with many other important areas of both pure and applied mathematics. The general goal of operator theory is to understand the structure of various classes of operators and collections of operators. The proposer plans to study this structure theory from various points of view. Since operators are infinite objects, in some sense it is impossible to understand completely what they are. It is therefore natural to study "approximations" of operators, just as we approximate numbers in terms of finitely many decimal places. Unlike the situation with numbers, where there is only one notion of "distance," there are many different notions of "closeness" for operators. The principal investigator plans to continue his extensive work on various approximation schemes, including certain distance formulas and a recent concept, free entropy, which is a type of approximation based on finite matrices. Another natural way to study operators is by breaking them into component parts and studying each of the parts separately. In finite dimensions this amounts to considering various decompositions of matrices, most notably the century-old Jordan canonical form. Unfortunately, it is very difficult to show that infinite-dimensional operators can always be broken into appropriate smaller parts. Indeed, this is the famous "invariant subspace problem," one of the most intriguing unsolved problems in all of mathematics. The proposer has new ideas for attacking this problem as well as the related problem of determining, given that an operator can be decomposed into smaller pieces, the extent to which every nearby operator is similarly decomposable.
