Cardinality of the set of topologically left invariant means on left uniformly continuous functionals

For a Lau algebra A with a certain condition we show that the number of topologically left invariant means on A is equal to the number of topologically left invariant means on LUC(A ). Using this for a nondiscrete locally comapct group G we prove that the cardinality of the set of topologically invariant means on UC(bG) is equal to 22b(G) . In particular, G is discrete if and only if there is a unique topologically invariant mean on UC(bG).

Keywords:

Fourier algebra , Lau algebra , left uniformly continuous functional , topologically left invariant mean.