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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The commutants of relatively prime powers in Banach algebras

Author: Abdullah H. Al-Moajil
Journal: Proc. Amer. Math. Soc. 57 (1976), 243-249
MSC: Primary 46K05; Secondary 47B99
MathSciNet review: 0407612
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Abstract: Let $ R$ be a ring and $ A(R) = \{ x \in R:x$ belongs to the second commutant of $ \{ {x^n},{x^{n + 1}}\} $ for all integers $ n > 1\} $. It is shown that in a prime ring $ R,A(R) = R$ if and only if $ R$ has no nilpotent elements. The set $ A(U)$ is studied for some special $ \ast $-algebras. It is shown that the normal elements of a proper $ \ast $-algebra $ U$ belong to $ A(U)$. If $ U$ is also prime then $ A(U) = \{ x \in U:x$ belongs to the second commutant of $ \{ {x^n},{x^{n + 1}}\} $ for some $ n > 1\} $. The set $ A(B(H))$ is studied, where $ B(H)$ is the algebra of bounded operators on a Hilbert space $ H$. Necessary and sufficient conditions for some special types of operators to belong to $ A(B(H))$ are obtained.

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

  • [1] A. H. Al-Moajil, Nilpotency and quasinilpotency in Banach algebras, Ph.D. Dissertation, University of Oregon, 1973.
  • [2] Sterling K. Berberian, Baer *-rings, Springer-Verlag, New York-Berlin, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 195. MR 0429975
  • [3] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part III, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral operators; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1971 original; A Wiley-Interscience Publication. MR 1009164
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Keywords: Double commutant, proper involution, prime, algebraic operator
Article copyright: © Copyright 1976 American Mathematical Society

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