Invariant Hermitian Kernels: A Unifying Approach
by
Aurelian Gheondea
Institute of Mathematics of the Romanian Academy

Given a set X and a Hilbert space H, a hermitian kernel is a mapping K:X×X --> L(H) such that K(x, y)=K(y, x)^* for all x, y in X. Assuming that S is a group and there is given an action \phi of S on the set X, the hermitian kernel is invariant if

K(\phi(a, x), \phi(a, y))=K(x, y),     x, y in X,  a in S.

We actually consider this definition in a more general context, when S is a semigroup with involution and there is given a \phi-multiplier \alpha that gives rise to a 2-cocycle \sigma. In this case, the definition of invariant hermitian kernel is generalized to that of projectively invariant hermitian kernel.

The problem we are considering refers to characterizing those projectively invariant hermitian kernels that produce projective representations of the semigroup with involution onto a Krein space. This problem is treated by means of Kolmogorov decompositions of hermitian kernels. Of special importance are the issues of similarity with representations on a Hilbert space and of uniqueness, modulo unitary equivalence.

This approach unifies results in the GNS-representations of hermitian maps on *-algebras, dilation theory associated with hermitian mappings of *-algebras, Wittstock decomposition of completely bounded maps on C^*-algebras and Weyl exponentials associated to an indefinite metric.

This report is based on joint work with Tiberiu Constantinescu.

Date received: May 24, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeo-29.