Topological invariant means on the von neumann algebra vn(g)

Let VN(G) be the von Neumann algebra generated by the left regular representation of a locally compact group G, A(G) the Fourier algebra of G and TIM(G) the set of topological invariant means on VN(G). Let (Formula Presanted) 0). We show that if G is nondiscrete then there exists a linear isometry A of (/°°)* into VN(G)* such that A(^, ) c TIM(G). When G is further assumed to be second countable then% can be embedded into some predescribed subsets of 77A/(G). To prove these embedding theorems for second countable groups we need the existence of a sequence of means(Formula Presented)in A(G) such that their supports in VN(G) are mutually orthogonal and Il uun - H J -* 0 if u is a mean in A(G). Let F(G) be the space of all T E VN(G) such that m(T) is a constant as m runs through TIM(G) and let W(G) be the space of weakly almost periodic elements in VN(G). We show that the following conditions are equivalent: (i) G is discrete, (ii) F(G) is an algebra and (iii) (Formula Presanted).