Continuity of translation invariant linear functionals on \(C_ 0(G)\) for certain locally compact groups G (Q1099027)

It is known that all translation invariant linear functionals on \(L_ p(G)\), \(1<p<\infty\), are continuous if G is not amenable [see the author, J. Aust. Math. Soc., Ser. A 41, 237-250 (1986; Zbl 0611.43001)] or if G is a compact group which has a dense subgroup with property T [\textit{J. Rosenblatt}, Proc. Am. Math. Soc. 94, 226-228 (1985; Zbl 0566.43001)]. It is shown in this paper that, in these cases, all translation invariant linear functionals on \(C_ 0(G)\) and \(L_{\infty}(G)\) are continuous as well. This follows from the elementary result that, if a representation of G on a Banach space, X, does not weakly contain the trivial representation, then there are no non-zero translation invariant linear functionals on \(X^*\), the dual space of X.