Unit Vectors, Morita Equivalence and Endomorphisms

TL;DR

This paper addresses fundamental questions in the theory of correspondences, establishing conditions for their representation as endomorphisms and on Hilbert spaces, with implications for product systems and W*-algebra representations.

Contribution

It provides solutions to key open problems about correspondences' representations and their relation to endomorphisms, extending the understanding of their structure and applications.

Findings

Every correspondence can be associated with a unital endomorphism of adjointable operators.

Every correspondence admits a nondegenerate faithful representation on a Hilbert space.

New proofs for classical results in W*-algebra representation theory.

Abstract

We solve two problems in the theory of correspondences that have important implications in the theory of product systems. The first problem is the question whether every correspondence is the correspondence associated (by the representation theory) with a unital endomorphism of the algebra of all adjointable operators on a Hilbert module. The second problem is the question whether every correspondence allows for a nondegenerate faithful representation on a Hilbert space. We also resolve an extension problem for representations of correspondences and we provide new efficient proofs of several well-known statements in the theory of representations of W*-algebras.