|Ph.D Student||Leonid Helmer|
|Subject||Factorizations and Reflexivity in Non Commutative Hardy|
|Department||Department of Mathematics||Supervisor||Full Professor Solel Baruch|
|Full Thesis text|
In 2004 P. Muhly and B. Solel introduced the non commutative Hardy algebras H∞(E), associated with a W* -correspondence E which generalize the classical Hardy algebra of the unit disc H∞ (D). As a special case one obtains also the algebra F∞ of Popescu, the free semigroup algebras, the quiver algebras and the analytic crossed products.
In the study of these non commutative analogs, one would like to understand to what extent known properties H∞ (D) can be generalized to properties of these non commutative algebras.
In this thesis we view the algebra H∞(E) as acting on a Hilbert space via an induced representation and write it ρ(H∞(E))and we study inner-outer factorizations and reflexivity for ρ(H∞(E)). Both issues were studied by Arias and Popescu in the context of the algebra F∞ generated by n shifts. But, as will be clear from our work, the extension to a more general W* -algebra M requires new techniques and approach. Key tools that we will need and use here is the version of Wold decomposition of a completely contractive representation of W*-correspondence E and the concept of duality for W *-correspondences.
We develop our version of the inner-outer factorization starting with the factorization of a vector on the underlying Hilbert space. We then construct the inner-outer factorization of an element of the commutant of our algebra, which is the Hardy algebra of the dual correspondence. Finally, using the duality of correspondences, we construct the factorization of an element of ρ(H∞(E)) which is natural for our version of inner-outer factorization.
Regarding reflexivity, we prove that every operator in Alg Lat ρ(H∞(E)) has a special matricial form. Then we take M to be a factor and under some additional assumptions we show that ρ(H∞(E)) is reflexive. By applying these results to the analytic crossed products ρ(H∞(αM)) we obtain their reflexivity for any automorphism α ∈ Aut(M) whenever M is a factor.
Finally, we show also the reflexivity of the compression of the Hardy algebra to a suitable coinvariant subspace M, which may be thought of as a generalized symmetric Fock space.