On Topologically Invariant Means on a Locally Compact Group

Let <!-- MATH $\mathcal{M}$ --> be the set of all probability measures on . Let G be a locally compact, noncompact, amenable group. Then there is a one-one affine mapping of <!-- MATH $\mathcal{M}$ --> into the set of all left invariant means on <!-- MATH ${L^\infty }(G)$ --> . Note that <!-- MATH $\mathcal{M}$ --> is a very big set. If we further assume G to be -compact, then we have a better result: The set <!-- MATH $\mathcal{M}$ --> can be embedded affinely into the set of two-sided topologically invariant means on <!-- MATH ${L^\infty }(G)$ --> . We also give a structure theorem for the set of all topologically left invariant means when G is -compact.