For a locally compact group G let \({\mathcal A}\) be the subset of functions f in \(L^{\infty}(G)\) which left average to some constant, i.e. some constant is in the \(\| \cdot \|_{\infty}\)-closure of convex combinations of left translates \({}_ xf\) of f. \({\mathcal A}_ 0\) is the set of functions which left average to 0 and H denotes \(span(\{f-_ xf:\) \(f\in L^{\infty}(G)\), \(x\in G\})\). It is shown that the following are equivalent: (i) G is amenable as a discrete group, (ii) \({\mathcal A}\) is a subspace of \(L^{\infty}(G)\), (iii) \({\mathcal A}_ 0\) is a subspace of \(L^{\infty}(G)\), (iv) \({\mathcal A}_ 0=\bar H.\)
A subspace of \(L^{\infty}(G)\) is called admissible if it contains constants and is left translation invariant. It is proved that amenability of G is equivalent to the existence of a largest admissible subspace with a unique left invariant mean (which is \(\bar H+{\mathbb{C}})\). A function f has a unique left invariant mean value if there exists a left invariant mean of the smallest admissible subspace generated by f and any such mean m satisfies \(m(f)=c\), c constant. It is shown that if the set of functions with a unique left invariant mean value is a subspace, then G is amenable. The converse holds for discrete groups. This answers problems asked by Emerson, Rosenblatt and Yang resp. Wong and Riazi.