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      Dilations of doubly invariant kernels valued in topologically ordered *- spaces

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      Embargo Lift Date: 2019-07-17
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      Author
      Ay, Serdar
      Advisor
      Gheondea E., Aurelian B. N.
      Date
      2018-07
      Publisher
      Bilkent University
      Language
      English
      Type
      Thesis
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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 *-Space
      VE-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/11693/47709
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      • Dept. of Mathematics - Ph.D. / Sc.D. 47
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