Todd Kemp

Todd Kemp
Ph.D. (2005) Cornell University

First Position

Escobar Assistant Professor, Cornell University


Hypercontractivity in Non-commutative Holomorphic Spaces


Research Area:
functional analysis, quantum probability theory

Abstract: In this dissertation, non-commutative holomorphic spacesHq (for –1 ≤ q ≤ 1) are introduced. These spaces naturally generalize the algebra of entire functions (identified with H1) in the context of the q-Gaussian von Neumann algebras Gammaq of Bozejko and Speicher. A non-commutative Segal-Bargmann transformSq — an isometric isomorphism L2(Gammaq)\to L2(Hq) which commutes with the number operator Nq, and which canonically generalizes the classical Segal-Bargmann transform — is constructed. The following theorem, which is the main result, is proved: for even integers r, the semigroup generated by Nq is a contraction L2(Hq)\to Lr(Hq) precisely when e–2t≤ 2/r. This strong hypercontractivity theorem generalizes (a special case of) the complex hypercontractivity result of Janson to the algebras Hq.