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 |