On topologically invariant means on a locally compact group and a conjecture of Paterson

John Hopfensperger

Published 2019-06-24Version 1

Let $G$ be a locally compact amenable group, and $\kappa$ be the smallest cardinality of a set of compacta covering $G$. Then $G$ has a Folner net of cardinality $\kappa$. We construct dense subsets of $TIM(G)$ and $TLIM(G)$ -- the topological (two-sided) invariant means and the topological left invariant means -- in terms of this Folner net. As an application, we give an elementary proof that $|TLIM(G)| = |TIM(G)| = 2^{2^\kappa}$. We also prove a conjecture of Paterson, that $TLIM(G) = TIM(G)$ iff $G$ has precompact conjugacy classes.