Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] William B. Arveson, Subalgebras of $ {C^\ast}$-algebras, Acta Math. 123 (1969), 141-224. MR 40 #6274. MR 0253059 (40:6274)
  • [2] -, Subalgebras of $ {C^\ast}$-algebras. II, Acta Math. 128 (1972), 271-308. MR 0394232 (52:15035)
  • [3] Robert R. Phelps, Lectures on Choquet's theorem, Van Nostrand, Princeton, N. J., 1966. MR 33 #1690. MR 0193470 (33:1690)
  • [4] Donald Sarason, On spectral sets having connected complement, Acta Sci. Math. (Szeged) 26 (1965), 289-299. MR 32 #6229. MR 0188797 (32:6229)
  • [5] W. F. Stinespring, Positive functions on $ {C^\ast}$-algebras, Proc. Amer. Math. Soc. 6 (1955), 211-216. MR 16, 1033. MR 0069403 (16:1033b)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46L05

Retrieve articles in all journals with MSC: 46L05


Additional Information

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

American Mathematical Society