Dilations of doubly invariant kernels valued in topologically ordered *- spaces
Embargo Lift Date: 2019-07-17
Gheondea E., Aurelian B. N.
Item Usage Stats
An ordered *-space Z is a complex vector space with a conjugate linear involution *, and a strict cone Z+ consisting of self adjoint elements. A topologically ordered *-space is an ordered *-space with a locally convex topology compatible with its natural ordering. A VE (Vector Euclidean) space, in the sense of Loynes, is a complex vector space equipped with an inner product taking values in an ordered *-space, and a VH (Vector Hilbert) space, in the sense of Loynes, is a VE-space with its inner product valued in a complete topologically ordered *-space and such that its induced locally convex topology is complete. On the other hand, dilation type theorems are important results that often realize a map valued in a certain space as a part of some simpler elements on a bigger space. Dilation results today are of an extraordinary large diversity and it is a natural question whether most of them can be uni*ed under general theorems. We study dilations of weakly positive semide*nite kernels valued in (topologically) ordered *-spaces, which are invariant under left actions of *-semigroups and right actions of semigroups, called doubly invariant. We obtain VE and VHspaces linearisations of such kernels, and on equal foot, their reproducing kernel spaces, and operator representations of the acting semigroups. The main results are used to unify many of the known dilation theorems for invariant positive semide*nite kernels with operator values, also for kernels valued in certain algebras, as well as to obtain some new dilation type results, in the context of Hilbert C*-modules, locally Hilbert C*-modules and VH-spaces.
KeywordsTopologically Ordered *-Space
Weakly Positive Semide nite Kernel
Doubly Invariant Kernel
Completely Positive Map
Showing items related by title, author, creator and subject.
Gheondea, A. (Springer, 2012)We investigate VH-spaces (Vector Hilbert spaces, or Loynes spaces) operator valued Hermitian kernels that are invariant under actions of *-semigroups from the point of view of generation of *-representations, linearizations ...
Constantinescu, T.; Gheondea, A. (Birkhaeuser Science, 2006)In previous works we analysed conditions for linearization of Hermitian kernels. The conditions on the kernel turned out to be of a type considered previously by L. Schwartz in the related matter of characterizing the real ...
Ay, Serdar; Gheondea, Aurelian (Instytut Matematyczny PAN, 2019)We consider weakly positive semidefinite kernels valued in ordered ∗-spaces with or without certain topological properties, and investigate their linearisations (Kolmogorov decompositions) as well as their reproducing ...