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Multipliers of group algebras of vector-valued functions

Authors: U. B. Tewari, M. Dutta and D. P. Vaidya
Journal: Proc. Amer. Math. Soc. 81 (1981), 223-229
MSC: Primary 43A22; Secondary 43A20
MathSciNet review: 593462
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Abstract: Let $ G$ be a locally compact abelian group and $ X$ be a Banach space. Let $ {L^1}(G,X)$ be the Banach space of $ X$-valued Bochner integrable functions on $ G$. We prove that the space of bounded linear translation invariant operators of $ {L^1}(G,X)$ can be identified with $ L(X,M(G,X))$, the space of bounded linear operators from $ X$ into $ M(G,X)$ where $ M(G,X)$ is the space of $ X$-valued regular, Borel measures of bounded variation on $ G$. Furthermore, if $ A$ is a commutative semisimple Banach algebra with identity of norm 1 then $ {L^1}(G,A)$ is a Banach algebra and we prove that the space of multipliers of $ {L^1}(G,A)$ is isometrically isomorphic to $ M(G,A)$. It also follows that if dimension of $ A$ is greater than one then there exist translationinvariant operators of $ {L^1}(G,A)$ which are not multipliers of $ {L^1}(G,A)$.

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Keywords: Locally compact abelian group, invariant operators, multiplier
Article copyright: © Copyright 1981 American Mathematical Society

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