Weak^∗-closed complemented invariant subspaces of L_∞(G) and amenable locally compact groups

Anthony To-Ming Lau and Viktor Losert

Vol. 123 (1986), No. 1, 149–159

DOI: 10.2140/pjm.1986.123.149

Abstract

One of the main results of this paper implies that a locally compact group G is amenable if and only if whenever X is a weak^∗-closed left translation invariant complemented subspace of L_∞(G), X is the range of a projection on L_∞(G) 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 ∈ G, on L₂(G), and Σ(M) = {g ∈ G;l_g ∈ 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 VN(G).

Mathematical Subject Classification 2000

Primary: 43A07

Secondary: 22D25

Milestones

Received: 10 January 1985

Published: 1 May 1986