### 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 (Fo -algebras for operator * -correspondences. We illustrate the relevance of this class of inner product modules by providing numerous examples arising from noncommutative geometry.

Original language | English |
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Journal | Proceedings of the London Mathematical Society |

Volume | 117 |

Issue number | 2 |

Pages (from-to) | 303-344 |

ISSN | 0024-6115 |

DOIs | |

Publication status | Published - 2018 |

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### Cite this

*Proceedings of the London Mathematical Society*,

*117*(2), 303-344. https://doi.org/10.1112/plms.12129