On the subalgebra of a Fourier–Stieltjes algebra generated by pure positive definite functions

For a locally compact group , the first-named author considered the closed subspace which is generated by the pure positive definite functions. In many cases is itself an algebra. We illustrate using Heisenberg groups and the real special linear group, that this is not the case in general. We examine the structures of the algebras thereby created and examine properties related to amenability.