Abstract
Let $T: A \longrightarrow B$ be a surjective operator between two unital semisimple commutative Banach algebras $A$ and $B$ with $T1=1$. We show that if $T$ satisfies the peripheral multiplicativity condition $\sigma_\pi(Tf.Tg) = \sigma_\pi (f.g)$ for all $f$ and $g$ in $A$, where $\sigma_\pi(f)$ shows the peripheral spectrum of $f$, then $T$ is a composition operator in modulus on the $\check{S}$ilov boundary of $A$ in the sense that $|f(x)|=|Tf(\tau(x))|,$ for each $f\in A$ and $x\in \partial(A)$ where $\tau: \partial (A) \longrightarrow \partial (B)$ is a homeomorphism between $\check{S}$ilov boundaries of $A$ and $B$.
Citation
M. Najafi tavani. "Peripherally multiplicative operators on unital commutative Banach algebras." Banach J. Math. Anal. 9 (3) 75 - 84, 2015. https://doi.org/10.15352/bjma/09-3-5
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