Multipliers of group algebras of vector-valued functions

Tewari, U. B.; Dutta, M.; Vaidya, D. P.

doi:10.1090/S0002-9939-1981-0593462-9

Multipliers of group algebras of vector-valued functions
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by U. B. Tewari, M. Dutta and D. P. Vaidya PDF

Proc. Amer. Math. Soc. 81 (1981), 223-229 Request permission

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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