Amenability of locally compact groups and subspaces of $L^ \infty (G)$

Abstract:

If $G$ is a locally compact group, let $\mathcal {A}$ be the set of all the functions which left average to a constant, i.e. the function $f \in {L^\infty }(G)$ such that there is a constant in the ${\left \| \right \|_{{\infty ^ - }}}$ closed convex hull of ${\{ _x}f:x \in G\}$. We prove in this paper that $\mathcal {A}$ is a subspace of ${L^\infty }(G)$ 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 $\mathcal {U}$ of functions in ${L^\infty }(G)$ admitting a unique left invariant mean value is a subspace of ${L^\infty }(G)$ when $G$ is not amenable and whether there is a largest admissible subspace of ${L^\infty }(G)$ with a unique left invariant mean.