A representation theorem for completely contractive dual Banach algebras and Connes-amenability

In the first part of the thesis, we prove a representation theorem for completely contractive dual Banach algebras. More explicitly, we prove that if A is a completely contractive dual Banach algebra, then there exists a w*-continuous complete isometry from A into CB (E), the operator space of completely bounded operators on E, for some reflexive operator space E.;In the second part of the thesis, we study the Connes-amenability of dual Banach algebras. We first prove some hereditary properties for the Connes-amenability. We present a necessary and sufficient condition for the Connes (and strongly Connes) amenability of w*-closed ideals of Connes (and strongly Connes) amenable dual Banach algebras. Every dual Banach algebra induces some short exact sequences. We characterize the Connes (and strongly Connes) amenability of dual Banach algebras in terms of certain homological properties of those short exact sequences.;Finally, we prove that for an arbitrary discrete group G, B( G), the Fourier-Stieltjes algebra of G is Connes-amenable if and only if G has an abelian subgroup of finite index.