GENERALIZED BOCHNER THEOREM AND LEVY-KHINCHINE REPRESENTATION FOR COMPLETELY POSITIVE-DEFINITE GENERALIZED TOEPLITZ KERNELS

The so-called generalized Bochner theorem, stated by Cotlar and Sadosk y, provides an integral representation of the positive definite genera lized Toeplitz kernels. In this paper we derive noncommutative analogu es of lifting theorems and the generalized Bochner theorem for complet ely positive definite generalized Toeplitz kernels, also considered by Cotlar and Sadosky, through the theory of unitary extensions of isome tric operators. Moreover, in the case where the kernel is defined in Z x Z, we can associate to each unitary extension an interpolation coll igation providing thus a wide class of liftings. Also a Levy-Khinchine type formula for this kind of kernels is obtained.