Properties of the set of topologically invariant means on P. Eymard's, W∗-algebra VN(G)

AbstractLet G be any locally compact group and VN(G) its associated Von Neumann algebra as in Eymard [3]. We show, for nondiscrete G, that the set of topologically invariant means (states) on VN(G) (denoted by TIM(G)) is not norm separable. In case G is second countable and nondiscrete, then TIM(G) does not have w∗ exposed points. These results improve a result of P. F. Renaud [7] who showed that VN(G) admits a unique topologically invariant mean if and only if G is discrete. We make at the end the conjecture which roughly states that for any nondiscrete G, TIM(G) does not possess w∗ exposed points. (For exact statement see end of paper.