Non-normal elements in Banach $^*$-algebras
Volume 160 / 2004
Studia Mathematica 160 (2004), 201-204
MSC: Primary 46L05.
DOI: 10.4064/sm160-3-1
Abstract
Let $A$ be a Banach $^*$-algebra with an identity, continuous involution, center $Z$ and set of self-adjoint elements ${\mit\Sigma}$. Let $h\in{\mit\Sigma}$. The set of $v\in{\mit\Sigma}$ such that $(h+iv)^n$ is normal for no positive integer $n$ is dense in ${\mit\Sigma}$ if and only if $h\not\in Z$. The case where $A$ has no identity is also treated.
