Dual Banach algebras associated to the coset space of a locally compact group

For H a closed group of a locally compact group G, B. Forrest ([4]) has defined the Fourier and Fourier Stieltjes algebras associated to the coset space G/H, A(G/H) and B(G/H) respectively. He proved that when H is compact, it is possible to extend many classical results to this new setting. We continue this investigation with the study of the dual space of A(G/H), showing that it can be identified with a particular type of w⁎-closed ideal of VN(G) determined by H. We obtain a characterization of all w⁎-closed left ideals of VN(G) that are of this particular form. We introduce and study the analogs of some of the classical subspaces of VN(G), UBC(Gˆ), W(Gˆ) and AP(Gˆ) in the new setting. We obtain results similar to those of D.E. Ramirez ([2]), E. Granirer ([6]) and A.T. Lau ([9]) obtained for these spaces in the group setting.