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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
DOI: https://doi.org/10.1090/S0002-9939-1976-0407612-6
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?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0407612-6
Keywords: Double commutant, proper involution, prime, algebraic operator
Article copyright: © Copyright 1976 American Mathematical Society