Multipliers on complemented Banach algebras

Tomiuk, Bohdan J.

doi:10.1090/S0002-9939-1992-1116273-1

Multipliers on complemented Banach algebras
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by Bohdan J. Tomiuk PDF

Proc. Amer. Math. Soc. 115 (1992), 397-404 Request permission

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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