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Multipliers on complemented Banach algebras


Author: Bohdan J. Tomiuk
Journal: Proc. Amer. Math. Soc. 115 (1992), 397-404
MSC: Primary 46H10
DOI: https://doi.org/10.1090/S0002-9939-1992-1116273-1
MathSciNet review: 1116273
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Abstract: Let $ A$ be a semisimple right complemented Banach algebra, $ {L_A}$ the left regular representation of $ A$, and $ {M_l}\left( A \right)$ the left multiplier algebra of $ A$. In this paper we are concerned with $ {L_A}$ and its relationship to $ A$ and $ {M_l}\left( A \right)$. We show that $ {L_A}$ is an annihilator algebra and that it is a closed ideal of $ {M_l}\left( A \right)$. Moreover, $ {L_A}$ and $ {M_l}\left( A \right)$ have the same socle. We also show that the left multiplier algebra of a minimal closed ideal of $ A$ is topologically algebra isomorphic to $ L\left( H \right)$, the algebra of bounded linear operators on a Hilbert space $ H$. Conditions are given under which $ {L_A}$ is right complemented.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1116273-1
Keywords: Right complemented Banach algebra, dual Banach algebra, left regular representation, left multiplier, abstract Segal algebra
Article copyright: © Copyright 1992 American Mathematical Society

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