Geometric Properties of Coefficient Function Spaces Determined by Unitary Representations of a Locally Compact Group

Let G be a locally compact group. As usual, let C*(G) denote the group C*-algebra of G and let B(G) denote its dual, the Fourier-Stieltjes algebra of G. If π is a (continuous) unitary representation of G, let Aπ be the space of coefficient functions of π and Bπ be its σ(B(G), C*(G))-completion in B(G). We first investigate the link between the Radon-Nikodým property for these coefficient function spaces and the complete reducibility of π. We then study the equivalence between the RNP and weak RNP for these spaces. In the next section, the Dunford-Pettis property for Bπ is characterised by the finite dimensionality of the irreducible representations weakly contained in π. The same characterisation holds for Aπ when VN(G), the group von Neumann algebra of G, has a nonzero type I, finite part. We note that the previous result cannot hold for all locally compact groups. We show finally that Aπ has the Schur property if and only if π is the direct sum of finite dimensional representations. At the end of each section, applications to the characterisation of certain classes of groups are given.