C*-algebras Generated by Fourier-stieltjes

For G a locally compact group and G its dual, let Jd(G) be the C*algebra generated by the Fourier-Stieltjes transforms of the discrete measures on G. We show that the canonical trace on ,4(G) is faithful if and only if G is amenable as a discrete group. We further show that if G is nondiscrete and amenable as a discrete group, then the only measures in dKd(G) are the discrete measures, and also the sup and lim sup norms are identical on ffd(G). These results are extensions of classical theorems on almost periodic functions on locally compact abelian groups. We present here some results dealing with nonabelian extensions of the theory of the von Neumann mean on the almost periodic functions. These results were previously announced in [3]. For G a locally compact group, let G be the dual of G. For _ G and ju E M(G), the measure algebra of G, let 7T(pL) be the Fourier-Stieltjes transform of j at i7. Let lISo be sup {-gr(k)l cT G }, and let &(G^) be the C*completion of M(G) relative to the norm 11 Let Ja(G) G4(O) be the closures in G(0) of L1(G) (the space of measures absolutely continuous with respect to left Haar measure), Md(G) (the space of discrete measures) respectively. The algebra G'(0) is a nonabelian analogue of the classical algebra of almost periodic functions. In ?1 we give some results dealing with the topology of G and also the spectrum of X~(G), denoted by KG. In the abelian case this is the closure of the dual group of G in the spectrum of M(G). We define the von Neumann trace on -&(G^) and derive some consequences. In ?2 we investigate the C*-extension of the canonical projection which maps a measure to its discrete part. This makes possible a characterization of the null space of the trace, and a proof that KG\G contains a homeomorphic copy of the reduced dual of Gd, the group G made discrete. In ?3 we show that the trace is faithful on 4d(G) if and only if Gd is amenable. We further show that if G is nondiscrete and Gd is amenable then the sup and lim sup norms are identical on #d(G(), and if pL E #d(G0) then u E Md(G). 1. For G a locally compact group, we write M(G) for the measure algebra of G, namely the set of finite regular Borel measures on G, Md(G) for the discrete Received by the editors August 28, 1970. AMS 1970 subject classifications. Primary 22D25, 43A25.