REPRESENTATIONS OF C∗-CORRESPONDENCES ON PAIRS OF HILBERT SPACES

We study representations of Hilbert bimodules on pairs of Hilbert spaces. If A is a C-& lowast;-algebra and X is a right Hilbert A-module, we use such representations to faithfully represent the C-& lowast;-algebras K-A(X) and L-A( X ) . We then extend this theory to define representations of ( A , B) ) C-& lowast;-correspondences on a pair of Hilbert spaces and show how these can be obtained from any nondegenerate representation of B . As an application of such representations, we give necessary and sufficient conditions on an ( A , B) ) C-& lowast;-correspondences to admit a Hilbert A- B-bimodule structure. Finally, we show how to represent the interior tensor product of two C-& lowast;-correspondences.