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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Boundary representations on $ C\sp{\ast} $-algebras with matrix units


Author: Alan Hopenwasser
Journal: Trans. Amer. Math. Soc. 177 (1973), 483-490
MSC: Primary 46L05
DOI: https://doi.org/10.1090/S0002-9947-1973-0322522-2
MathSciNet review: 0322522
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Abstract: Let $ \mathcal{A}$ be a $ {C^\ast}$-algebra with unit, let $ \mathcal{S}$ be a linear subspace of $ \mathcal{A} \otimes {M_n}$ which contains the natural set of matrix units and which generates $ \mathcal{A}$ as a $ {C^\ast}$-algebra. Let $ \mathcal{J}$ be the subset of $ \mathcal{A}$ consisting of entries of matrices in $ \mathcal{S}$. Then the boundary representations of $ \mathcal{A} \otimes {M_n}$ relative to $ \mathcal{S}$ are parametrized by the boundary representations of $ \mathcal{A}$ relative to $ \mathcal{J}$. Also, a nontrivial example is given of a subalgebra of a $ {C^\ast}$-algebra which possesses exactly one boundary representation.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0322522-2
Keywords: $ {C^\ast}$-algebra, boundary representation, matrix units, Choquet boundary
Article copyright: © Copyright 1973 American Mathematical Society