Representations of $\ast$-semigroups associated to invariant kernels with values adjointable operators. I

Abstract

We consider positive semidefinite kernels valued in the $\ast$-algebra of adjointable operators on a VE-space (Vector Euclidean space) and that are invariant under actions of $\ast$-semigroups. A rather general dilation theorem is stated and proved: for these kind of kernels, representations of the $\ast$-semigroup on either the VE-spaces of linearisation of the kernels or on their reproducing kernel VE-spaces are obtainable. We point out the reproducing kernel fabric of dilation theory and we show that the general theorem unifies many dilation results at the non topological level.

23 pages

References in corpus (1)

• Arveson's extension theorem in *-algebras

Cited by in corpus (3)

• Representations of *-semigroups associated to invariant kernels with values continuously adjointable operators
• Isometric Representations of Calibrated Ordered Spaces on $C(X)$
• Automatic Boundedness of Adjointable Operators on Barreled VH-Spaces