Asymptotics of Matrix Coefficients and Closures of Fourier–Stieltjes Algebras

The vanishing theorems of R. Howe and C. C. Moore are interpreted in a topological way. This yields a generalization of their results to connected locally compact totally minimal groups. The first section gives the complete classification of these groups and shows a lot of examples. It is proven that, among the connected groups, the totally minimal groups are exactly those groupsGwhose Fourier–Stieltjes algebras are uniformly dense in the C*-algebra of all weakly almost periodic functions WAP(G). This solves a problem which was intensely discussed by C. Chou. Furthermore it is shown that WAP(G) is generated as a C*-algebra by the continuous functions onG/Nwhich vanish at infinity, whereNranges over all closed normal subgroups.