Dilations of doubly invariant kernels valued in topologically ordered *- spaces
Author
Ay, Serdar
Advisor
Gheondea E., Aurelian B. N.
Date
2018-07Publisher
Bilkent University
Language
English
Type
ThesisItem Usage Stats
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Abstract
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.
Keywords
Topologically Ordered *-SpaceVE-Space
VH-Space
Hermitian Kernel
Weakly Positive Semide nite Kernel
Doubly Invariant Kernel
Linearisation
Reproducing Kernel
*-representation
Completely Positive Map
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http://hdl.handle.net/11693/47709Collections
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