Compactness of a locally compact group G and geometric properties of A(p)(G)

Let G be a locally compact topological group. A number of characterizations are given of the class of compact groups in terms of the geometric properties such as Radon-Nikodym property, Dunford-Pettis property and Schur property of A(p)(G), and the properties of the multiplication operator on PFp(G). We extend and improve several results of Lau and Ulger [17] to A(p)(G) and B-p(G) for arbitrary p.