Isometric multipliers and isometric isomorphisms of $l_{1}(S)$

Abstract:

Let $S$ be a commutative semigroup and $\Omega (S)$) the multiplier semigroup of $S$. It is shown that $T$ is an isometric multiplier of ${l_1}(S)$ if and only if there exists an invertible element $\sigma \in \Omega (S)$ and a complex number $\lambda$ of unit modulus such that $T(\alpha ) = \lambda \sum \nolimits _{x \in S} {\alpha (x){\delta _{\sigma (x)}}}$ for each $\alpha = \sum \nolimits _{x \in S} {\alpha (x){\delta _x} \in {l_1}} (S)$. Also, if ${S_1}$ and ${S_2}$ are commutative semigroups, and $L$ is an isometric isomorphism of ${l_1}({S_1})$ into ${l_1}({S_2})$, then it is proved that there exist a semicharacter $\chi ,|\chi (x)| = 1$ for all $x \in {S_1}$, and an isomorphism $i$ of ${S_1}$ onto ${S_2}$ such that $L(\alpha ) = \sum {\chi (x)\alpha (x){\delta _{i(x)}}}$ for each $\alpha = \sum \nolimits _{x \in {S_1}} {\alpha (x){\delta _x}} \in {l_1}({S_1})$.