On functional calculus for Hermitian elements of Banach algebras: the norm and spectral radius

Abstract

Let $A$ be a complex unital Banach algebra. An element $a\in A$ is said to be Hermitian, if $\Vert \exp (ita)\Vert =1$ for all $t\in R$. In the case of the algebra of bounded linear operators in a Hilbert space this Hermitian property agrees with the ordinary selfadjointness. If $a\in A$ is Hermitian, then $|a|=||a||$, where $|a|$ denotes the spectral radius of $a$. A function $F: R\to \C$ is called the universal symbol if $\Vert |F(a)||=|F(a)|$\ for each $A$ and all Hermitian $a\in A$. We characterize universal symbols in terms of positive definite functions.

10 pages