The commutants of relatively prime powers in Banach algebras
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- by Abdullah H. Al-Moajil PDF
- Proc. Amer. Math. Soc. 57 (1976), 243-249 Request permission
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
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A. H. Al-Moajil, Nilpotency and quasinilpotency in Banach algebras, Ph.D. Dissertation, University of Oregon, 1973.
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- Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
Additional Information
- © Copyright 1976 American Mathematical Society
- 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