Isometric multipliers of a vector valued Beurling algebra on a discrete semigroup
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Let $(S, \omega)$ be a weighted abelian semigroup, let $M_\omega(S)$ be the semigroup of $\omega$-bounded multipliers of $S$, and let $\mathcal{A}$ be a strictly convex commutative Banach algebra with identity. It is shown that $T$ is an onto isometric multiplier of $\mathcal{l}^1(S, \omega, \mathcal{A})$ if and only if there exists an invertible $\sigma\in M_\omega(S)$, a unitary point $a\in\mathcal{A}$, and a $k > 0$ such that $T(f) = ka\sum_{x\in S}f(x)\delta_{\sigma (x)}$ for each $f = \sum_{x\in S}f(x)\delta_x\in\mathcal{l}^1(S, \omega, \mathcal{A})$. It is also shown that an isomorphism from $\mathcal{l}^1(S_1, \omega_1, \mathcal{A})$ onto $\mathcal{l}^1(S_2, \omega_2, \mathcal{B})$ induces an isomorphism from $M(\mathcal{l}^1(S_1, \omega_1, \mathcal{A}))$, the set of all multipliers of $\mathcal{l}^1(S_1, \omega_1, \mathcal{A})$, onto $M(\mathcal{l}^1(S_2, \omega_2, \mathcal{B}))$.
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Volume 129 | Issue 3
June 2019
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