Operator -correspondences in analysis and geometry

Abstract

An operator -algebra is a non-self-adjoint operator algebra with completely isometric involution. We show that any operator -algebra admits a faithful representation on a Hilbert space in such a way that the involution coincides with the operator adjoint up to conjugation by a symmetry. We introduce operator -correspondences as a general class of inner product modules over operator -algebras and prove a similar representation theorem for them. From this, we derive the existence of linking operator -algebras for operator -correspondences. We illustrate the relevance of this class of inner product modules by providing numerous examples arising from noncommutative geometry.