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Algebraically admissible cones in free products of $*$-algebras


It was proved in~\cite{Pop09b} that a $*$-algebra is $C^*$-representable, i.e., $*$-isomorphic to a self-adjoint subalgebra of bounded operators acting on a Hilbert space if and only if there is an algebraically admissible cone in the real space of Hermitian elements of the algebra such that the algebra unit is an Archimedean order unit. In the present paper we construct such cones in free products of $C^*$-representable $*$-algebras generated by unitaries. We also express the reducing ideal of any algebraically bounded $*$-algebra with corepresentation $\mathcal F/\mathcal J$ where $\mathcal F$ is a free algebra as a closure of the ideal $\mathcal J$ in some universal enveloping $C^*$-algebra.

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TitleAlgebraically admissible cones in free products of $*$-algebras
SourceMethods Funct. Anal. Topology, Vol. 16 (2010), no. 1, 51-56
MathSciNet   MR2656131
CopyrightThe Author(s) 2010 (CC BY-SA)

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Stanislav Popovych
Kyiv National Taras Shevchenko University, Mechanics and Mathematics Department, acad. Glushkova 6, Kyiv, 02127, Ukraine 

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Stanislav Popovych, Algebraically admissible cones in free products of $*$-algebras, Methods Funct. Anal. Topology 16 (2010), no. 1, 51-56.


@article {MFAT534,
    AUTHOR = {Popovych, Stanislav},
     TITLE = {Algebraically admissible cones in free products of $*$-algebras},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {16},
      YEAR = {2010},
    NUMBER = {1},
     PAGES = {51-56},
      ISSN = {1029-3531},
  MRNUMBER = {MR2656131},
       URL = {},

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