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 closedL1 (G)-invariant complement inL∞ (G); (iii) the annihilatorX⊥ ofX inL1 (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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