Operator *-correspondences in analysis and geometry

David Blecher, Jens Kaad, Bram Mesland

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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 languageEnglish
JournalProceedings of the London Mathematical Society
Volume117
Issue number2
Pages (from-to)303-344
ISSN0024-6115
DOIs
Publication statusPublished - 2018

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