On rings and Banach algebras with skew derivations

Abstract

In the present paper, we investigate the commutativity of a prime Banach algebra with skew derivations and prove that if $\Aa$ is prime Banach algebra and $\Aa$ has a nonzero continuous linear skew derivation $\f$ from $\Aa$ to $\Aa$ such that $[\f(\xa^{m}), \f(\ya^{n})] - [\xa^{m}, \ya^{n}] \in \z(\Aa)$ for an integers $m = m(\xa, \ya)>1$ and $n = n(\xa, \ya)>1$ and sufficiently many $\xa, \ya$, then $\Aa$ is commutative.