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 is similar to a -representation on .

**[1]**Sandra Barkdull Cleveland,*Homomorphisms of non-commutative *-algebras*, Pacific J. Math.**13**(1963), 1097–1109. MR**0158274****[2]**Nathan Jacobson,*Structure of rings*, American Mathematical Society, Colloquium Publications, vol. 37, American Mathematical Society, 190 Hope Street, Prov., R. I., 1956. MR**0081264****[3]**Richard V. Kadison,*On the orthogonalization of operator representations*, Amer. J. Math.**77**(1955), 600–620. MR**0072442****[4]**Irving Kaplansky,*The structure of certain operator algebras*, Trans. Amer. Math. Soc.**70**(1951), 219–255. MR**0042066**, 10.1090/S0002-9947-1951-0042066-0**[5]**Charles E. Rickart,*General theory of Banach algebras*, The University Series in Higher Mathematics, D. van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0115101**

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DOI:
https://doi.org/10.1090/S0002-9939-1971-0290118-0

Keywords:
Algebras of operators,
irreducible algebra,
irreducible representation,
-algebra

Article copyright:
© Copyright 1971
American Mathematical Society