Weak^∗-closed complemented invariant subspaces of<i>L</i>_∞(<i>G</i>) and amenable locally compact groups

Abstract

One of the main results of this paper implies that a locally compact group G is amenable if and only ii whenever X is a weak*-closed left translation invariant complemented subspace of L^iG), X is the range of a projection on L^iG) commuting with left translations. We also prove that if G is a locally compact group and M is an invariant W^-subalgebra of the von Neumann algebra VN(G) generated by the left translation operators l g , g e G, on L 2 (G), and (M) {g e G; g e M} is a normal subgroup of G, then M is the range of a projection on VN(G) commuting with the action of the Fourier algebra A(G) on l Introduction. Let G be a locally compact group and L^iG) be the algebra of essentially bounded measurable complex-valued functions on G with pointwise operations and essential sup norm. Let X be a weak*-closed left translation invariant subspace of L^G). Then X is invaantly complemented in L^G) if X admits a left translation invariant closed complement, or equivalently, X is the range of a continuous projection on L^G) commuting with left translations.