Criteria for compactness and for discreteness of locally compact amenable groups

Abstract:

Let $G$ be a locally compact group $P(G) = \{ 0 \leqq \phi \in {L_1}(G);\int {\phi (x)dx = 1\} }$ and $({l_a}f)(x) = {}_af(x) = f(ax)$ for all $a,x \in G$ and $f \in {L^\infty }(G).0 \leqq \Psi \in {L^\infty }{(G)^ \ast },\Psi (1) = 1$ is said to be a [topological] left invariant mean ([TLIM] LIM) if $\Psi {{\text {(}}_a}f) = \Psi (f)[\Psi (\phi \ast f) = \Psi (f)$] for all $a \in G,\phi \in P(G),f \in {L^\infty }(G)$. The main result of this paper is the Theorem. Let $G$ be a locally compact group, amenable as a discrete group. If $G$ contains an open $\sigma$-compact normal subgroup, then LIM = TLIM if and only if $G$ is discrete. In particular if $G$ is an infinite compact amenable as discrete group then there exists some $\Psi \in LIM$ which is different from normalized Haar measure. A harmonic analysis type interpretation of this and related results are given at the end of this paper.$^{2}$