Operator algebras and operator theory

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This proposal concerns the study of continuous linear maps on Hilbert space (operators) and the algebras that they generate (operator algebras). We are seeking interesting structural properties of operator algebras that reveal their inner workings. Generally we wish to relate analytic invariants with algebraic or combinatorial invariants of some underlying object associated to the algebra. We plan to build on some of our notable recent successes. The hope and expectation is to make significant progress, and make important contributions to the discipline. Our recent work has had a strong influence on the field, and we are well positioned to continue to have a significant impact.Multivariable operator theory seeks to study a (finite) set of (usually commuting) operators. The algebra of such a set is well developed, but the analysis is in a more rudimentary stage. We have established a strong functional calculus for such sets, and this should lead to powerful new methods. We are interested in finding invariant subspaces (triangular forms). There is a very interesting ideal structure in the universal algebra for these sets of operators, and we seek to refine our earlier analysis. This leads to questions about interpolation of given function on some small set with constraints on the norm.Non-commutative convexity seeks to generalize ideas from classical convexity theory and approximation theory to the operator context. A matrix convex set has additional structure associated to higher dimensions. A famous conjecture of Arveson lays out a very interesting question related to approximation theory. We have answered this question in the commutative setting, which led to new developments in the classical theory and stronger approximation results. We hope to extend this to the non-commutative case.Associated to any directed graph, there are several operator algebras. These are often studied via the Cuntz-Kreiger C*-algebra. Here we instead study the weakly closed nonself-adjoint operator algebra. This leads to interesting structure and many questions. Our results relate back to invariants for the representations of the C*-algebra. We are currently working on a quantitative version of reflexivity known as hyper-reflexivity. This has been established for free semigroup algebras, which is the case of a graph with one vertex, and we have strong reasons to believe that it will follow in general.We have studied the problem of isomorphism between algebras associated to varieties on the complex ball. We have been successful for homogeneous varieties, but in the general case, there are many obstacles. We are seeking new geometric invariants that will provide new information. In particular, we are trying to show that if the varieties are suitably close, then their multiplier algebras are spatially equivalent (similar).