On the subalgebra of a Fourier-Steiltjes algebra generated by pure positive definite functions

For a locally compact group $G$, the first-named author considered the closed subspace $a_0(G)$ which is generated by the pure positive definite functions. In many cases $a_0(G)$ is itself an algebra. We illustrate using Heisenburg groups and the $2\times 2$ 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.