The commutants of relatively prime powers in Banach algebras

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.