Operator *-correspondences in analysis and geometry

David Blecher, Jens Kaad, Bram Mesland

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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 languageEnglish
JournalProceedings of the London Mathematical Society
Issue number2
Pages (from-to)303-344
Publication statusPublished - 2018


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