Title:

On nonabelian harmonic analysis and compactness conditions

In this thesis we study problems of Harmonic analysis type on locally compact not necessarily abelian groups. Let G be such a group. The Fourier Stieltjes algebra B(G) and the Fourier algebra A(G) are known to generalize the measure algebra M(G) and the group algebra L1(G) of the dual group G of a locally compact abelian group G. We show that Ginvariant subspaces of B(G) are the right generalizations of Lsubspaces of M(G), every Ginvariant subalgebra of B(G) is a predual of a Hopfvon Neumann algebra and the maximal ideal space of a Ginvariant subalgebra is a semigroup. We prove the following characterization theorem for A(G): "Let A be a selfadjoint Ginvariant subalgebra of B(G) such that (a) AA = G and (b) An A(G) ≠0 then (c) A(G) c A c A1/2(G)", where A1/2(G) is the radical of A(G) in B(G). We give a list of groups for which condition (b) is automatically satisfied for any selfadjoint Ginvariant subalgebra A that has condition (a) and we give characterizations for the Ginvariant ideals B0(G) and A1/2(G) by a continuity property of translations. More can be said about the algebras B(G) and A(G) if the groups in study are not merely locally compact. We study groups that have compactness conditions on them. We show that the Silov boundary δB(G) is not the whole of the maximal ideal space ΔB(G) and that B(G) is an asymmetric algebra for a class of groups that contain noncompact locally compact abelian groups and nonperiodic FC]groups. We study the commutative Banach algebra ZA(G) of central functions in A(G). We show that ZA(G) is isometric and isomorphic to an algebra L1(X,m) where X is a commutative hypergroup and m is an invariant measure on X.
