The GBT's lifting approach is valid in abstract algebraic and hilbertian scattering systems, with one or several evolution groups (not necessarily commuting), and integral representations of Toeplitz extensions of Hankel forms are obtained in many such systems. These integral representations lead to applications to harmonic analysis in product spaces, such as the polydisk, and in symplectic spaces. In a different direction, a noncommutative extension of the GBT is given for kernels denned in terms of completely positive maps.

Introduction

The study of the generalized Toeplitz kernels and forms started as an attempt to apply Krem's moment theory methods to the Hubert transform. In particular, a generalization of the classical Herglotz-Bochner theorem, the GBT, yields a characterization of the pairs of measures for which the Hubert transform operator is continuous in the corresponding weighted L2 spaces. Yet the GBT, unlike the Bochner theorem, provides not only integral representations of bounded forms, but invariant extensions of them without norm increase. The GBT, therefore, isclosely related with the far-reaching lifting theory of Sz.-Nagy and Foias: their lifting theorem for intertwining contractions can be obtained as a corollary of its abstract generalization to scattering systems (see Section 2 below).