Ph.D Thesis

Ph.D StudentHelmer Leonid
SubjectFactorizations and Reflexivity in Non Commutative Hardy
DepartmentDepartment of Mathematics
Supervisor PROFESSOR EMERITUS Baruch Solel
Full Thesis textFull thesis text - English Version


  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.