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Isometric multipliers and isometric isomorphisms of $ l\sb{1}(S)$


Author: Charles D. Lahr
Journal: Proc. Amer. Math. Soc. 58 (1976), 104-108
MSC: Primary 43A22
DOI: https://doi.org/10.1090/S0002-9939-1976-0415209-7
MathSciNet review: 0415209
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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 ,\vert\chi (x)\vert = 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})$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0415209-7
Keywords: Isometric multiplier, $ {l_1}$-algebra
Article copyright: © Copyright 1976 American Mathematical Society

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