# Operator *-correspondences in analysis and geometry

## TL;DR

This paper develops a framework for operator *-algebras and *-correspondences, providing representation theorems and linking structures, with applications to noncommutative geometry.

## Contribution

It introduces operator *-correspondences and proves their faithful representation theorem, extending the theory of operator algebras in analysis and geometry.

## Key findings

- Existence of faithful representations for operator *-algebras.
- Introduction of operator *-correspondences as inner product modules.
- Construction of linking operator *-algebras for these correspondences.

## Abstract

An operator *-algebra is a non-selfadjoint 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.