Almost periodic operators in vn(g)

Let G be a locally compact group, A(G) the Fourier algebra of G, B(G) the Fourier-Stieltjes algebra of G and VN(G) the von Neumann algebra generated by the left regular representationλ of G.Then A(G) is the predualof VN(G); VN(G) is a B(G)-module and A(G) is a closed ideal of B(G).Letis a compact operator from A(G) into VN(G)), the space of almost periodic operators inVN(G).Letbe the C*-algebra generated by (λ(x): x ∈ G). ThenFor a compact G, let E be the rank one operator on L²(G) that sends h∈ L²(G)to the constant function ∫ h(x) dx. We have the following results: (I) There exists a compact group G such that2) For a compact Lie group Ghas a unique left invariant mean ⇔G is semisimple. (3) If G is an extension of a locally compact abelian group by an amenable discrete group then(4) Let G = F_r, the free group with r generators, I <r < ∞. If T ∈ VN(G)and is a compact operator from B(G)into VN(G).