Operator ∗‐correspondences in analysis and geometry

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 ∗‐algebras for operator ∗‐correspondences. We illustrate the relevance of this class of inner product modules by providing numerous examples arising from noncommutative geometry.

Curvature of differentiable Hilbert modules and Kasparov modules

“…This gives the completion B 2 of B in the norm • 2 the structure of an operator *algebra [3]. By construction, the inclusions B 2 → B 1 → B are completely contractive operator * -algebra homomorphisms.…”

“…We carefully capture the corresponding C 1 -and C 2 -topologies in terms of suitable operator * -algebras (see [3]), which we will first introduce.…”

“…We write B for the C * -closure of B in the norm it inherits as an algebra of operators on B(Y ). An operator space is a closed subspace of B(H), an operator algebra is an operator space that is closed under multiplication in B(H), and an operator * -algebra is an operator algebra that carries a completely isometric involution, [3,Definition 1.4].…”

Involutive operator algebras

“…Examples from noncommutative differential geometry. Several examples of operator * -algebras were given in [10], most of them examples from noncommutative differential geometry (historically the first such example being due to Mesland). Other examples from noncommutative differential geometry may be found in other recent papers of Kaad, Mesland, and their coauthors.…”

“…Examples include the operator * -algebras occurring in noncommutative differential geometry studied recently by Mesland, Kaad, Lesch, and others (see e.g. [26,25,10] and references therein), (complexifications) of real operator algebras, and an operator algebraic version of the complex symmetric operators studied by Garcia, Putinar, Wogen, Zhu, and many others (see [21] for a survey, or e.g. [22]).…”