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For G an infinite compact group we show the existence and nonuniqueness of invariant means on the dual of the Fourier algebra. It follows that the space of weakly almost periodic functionals on the Fourier algebra is a proper closed subspace of the dual of the Fourier algebra. We let G denote an infinite compact group and G its dual. We use the notation of [2, Chapters 7 and 8]. Recall A(G) denotes the Fourier algebra of G (an algebra of continuous functions on G) and =Sf °°(G) denotes its dual under the pairing (/, ) (feA(G), $£^{0)). Further note a"°{u) is identified with the C*-algebra of bounded operators on L2(G) commuting with right translation. The module action of A(G) on ??""{o) is defined by the following: for feA(G), e^°°(G), fi 0, (2)p(I)=l (7 is the identity in J^°°(G)), and (3) p(f- )=f(e)p{ || = 1.) If G is also abelian, then G is a discrete group and (3) is equivalent to p( ), by x. Let P={feA(G):f(e)=\ and/is positive definite}. Now £ is a convex spanning subset of A(G) and a commutative semigroup under pointwise multiplication. Further P contains functions with supports in arbitrarily small neighborhoods of e. The collection of invariant means on J^X(G) will be denoted by Jf. Thus for pe^x(G)* (the dual of -2"° ((?)), peJT if and only if (l)p(cp)>0 whenever c^O, (2)p(I)=1, and {3)p(f- ),feP, -/(«), and f-I=f(e)I (feA(G)). In JS?°°(G) let iv* denote the weak-* topology oLx denote the canonical embedding of A(G) intoi?10^)*. Received by the editors October 1, 1970 and, in revised form, October 29, 1970. AMS 1970 subject classifications. Primary 43A75, 43-00.

Existence and nonuniqueness of invariant means on $\mathcal{L}^\infty(\hat G)$