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Title: On non-abelian harmonic analysis and compactness conditions
Author: Albar, S. F.
Awarding Body: University of Aberdeen
Current Institution: University of Aberdeen
Date of Award: 1981
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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 G-invariant subspaces of B(G) are the right generalizations of L-subspaces of M(G), every G-invariant subalgebra of B(G) is a predual of a Hopf-von Neumann algebra and the maximal ideal space of a G-invariant subalgebra is a semigroup. We prove the following characterization theorem for A(G): "Let A be a self-adjoint G-invariant 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 self-adjoint G-invariant subalgebra A that has condition (a) and we give characterizations for the G-invariant 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 non-compact locally compact abelian groups and non-periodic 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 hyper-group and m is an invariant measure on X.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available