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Irreducible algebras of operators which contain a minimal idempotent.

Author: Bruce A. Barnes
Journal: Proc. Amer. Math. Soc. 30 (1971), 337-342
MSC: Primary 46.65
MathSciNet review: 0290118
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Abstract: We prove that when A is a closed subalgebra of the bounded operators on a reflexive Banach space X, which acts irreducibly on X and contains a minimal idempotent, then every bounded operator with finite dimensional range on X is in A. We use this result to prove that every continuous irreducible representation of a GCR-algebra on a Hilbert space $ \mathcal{H}$ is similar to a $ ^ \ast $-representation on $ \mathcal{H}$.

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

  • [1] S. B. Cleveland, Homomorphisms of non-commutative $ ^ \ast $-algebras, Pacific J. Math. 13 (1963), 1097-1109. MR 28 #1500. MR 0158274 (28:1500)
  • [2] N. Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R. I., 1956. MR 18, 373. MR 0081264 (18:373d)
  • [3] R. V. Kadison, On the orthogonalization of operator representations, Amer. J. Math. 77 (1955), 600-620. MR 17, 285. MR 0072442 (17:285c)
  • [4] I. Kaplansky, The structure of certain operator algebras, Trans. Amer. Math. Soc. 70 (1951), 219-255. MR 13, 48. MR 0042066 (13:48a)
  • [5] C. E. Rickart, Banach algebras, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #5903. MR 0115101 (22:5903)

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Keywords: Algebras of operators, irreducible algebra, irreducible representation, $ {B^ \ast }$-algebra
Article copyright: © Copyright 1971 American Mathematical Society

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