Mathematical Sciences: Noncommutative Harmonic Analysis and Operator Algebras

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9431954 POPESCU This proposal considers some problems in ``Noncommutative harmonic analysis and operator algebras''. The framework of this proposal is mainly the C*-algebra (resp. non-self-adjoint algebra) generated by a class of semigroups of isometries on Hilbert spaces. Noncommutative Poisson transforms, Wold-type decompositions, positive-definite functions on semigroups, and isometric dilations are considered to study these algebras and their representations. The proposer expects to find noncommutative analogs to some classical results. The main directions of this proposed research are the following: 1. Poisson transforms on some C*-algebras generated by isometries. 2. Joint dilations and universal operator algebras. 3. Positive-definite functions on semigroups and isometric representations. 4. Noncommutative analog of the Hardy space of bounded analytic functions on the unit disk. The motivation of this research is the recent worldwide interest in the noncommutative aspect of harmonic analysis originated from the concept of quantization which links together several branches of mathematics and is closely related to mathematical physics. The objective of this research is to advance the understanding of this relatively new area of mathematics and to apply some of these results to noncommutative probability theory, interpolation theory, and model theory.