AMENABILITY OF LOCALLY COMPACT-GROUPS AND SUBSPACES OF L-INFINITY (G)

If G is a locally compact group, let A be the set of all the functions which left average to a constant, i.e. the function f epsilon L-infinity (G) such that there is a constant in the parallel-to.parallel-to-infinity- closed convex hull of {x-f:x epsilon G}. We prove in this paper that A is a subspace of L infinity (G) if and only if G is amenable as a discrete group. This answers a problem asked by Emerson, Rosenblatt and Yang, and Wong and Riazi. We also answer two other problems of Rosenblatt and Yang on whether the set U of functions in L-infinity (G) admitting a unique left invariant mean value is a subspace of L-infinity (G) G is not amenable and whether there is a largest admissible subspace of L-infinity (G) with a unique left invariant mean.