Todd Kemp
Todd Kemp

Ph.D. (2005) Cornell University

First Position
Dissertation
Advisor:
Research Area:
Abstract: In this dissertation, noncommutative holomorphic spacesH_{q} (for –1 ≤ q ≤ 1) are introduced. These spaces naturally generalize the algebra of entire functions (identified with H_{1}) in the context of the qGaussian von Neumann algebras Gamma_{q} of Bozejko and Speicher. A noncommutative SegalBargmann transformS_{q} — an isometric isomorphism L^{2}(Gamma_{q})\to L^{2}(H_{q}) which commutes with the number operator N_{q}, and which canonically generalizes the classical SegalBargmann transform — is constructed. The following theorem, which is the main result, is proved: for even integers r, the semigroup generated by N_{q} is a contraction L^{2}(H_{q})\to L^{r}(H_{q}) precisely when e^{–2t}≤ 2/r. This strong hypercontractivity theorem generalizes (a special case of) the complex hypercontractivity result of Janson to the algebras H_{q}.