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Complemented subspaces ofL ∞ (G), ideals ofL 1 (G) and amenability

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Abstract

LetG be a locally compact group andX a weak *-closed translation invariant subspace ofL ∞ (G). It is shown that the following conditions are equivalent: (i)X has a closedG-invariant complement inL ∞ (G); (ii)X has a closedL 1 (G)-invariant complement inL ∞ (G); (iii) the annihilatorX ⊥ ofX inL 1 (G) has bounded approximate units. The following result of Lau and Losert is then deduced: ifG is amenable andX complemented, thenX has a closedG-invariant complement. This implies for amenableG thatX is complemented if and only if the idealX ⊥ has bounded approximate units. This duality unifies and generalizes results of Gilbert, Liu, van Rooij, Wang, Rosenthal and Reiter.

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  1. Mathematisches Institut, Technische Universität München, Arcisstrasse 21, D-8000, München 2, Germany

    Mohammed E. B. Bekka

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  1. Mohammed E. B. Bekka
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Bekka, M.E.B. Complemented subspaces ofL ∞ (G), ideals ofL 1 (G) and amenability. Monatshefte für Mathematik 109, 195–203 (1990). https://doi.org/10.1007/BF01297760

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  • DOI: https://doi.org/10.1007/BF01297760

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