A characterization of quotient algebras of Ll(G)

Donald E. Ramirez
1969 Mathematical proceedings of the Cambridge Philosophical Society (Print)  
Let G be a locally compact Abelian group; I' the dual group of G; CO(l_')the algebra of continuous functions on r which vanish at infinity; CB(17) the continuous, bounded functions on r; M(G) the algebra of bounded Borel measures on G; Ll(G) the algebra of absolutely continuous measures; and M(G)A the algebra of l?ourier-Stielt jes transforms. Let E denote a compact subset of I', and let A(E) be the set of all functions on E which are restrictions to E of functions belonging to L1(G)A. The norm
more » ... in A(E) is the quotient norm; i.e. llfll~(~)= 'nf{llg\l~l(Gi gGL'(G) and @)E 'f} For f e C(E), let and IlfllCms llill~m)s 11.fllAE)) for f ~A(E). THEOREM 1. B(E) i8 a commutative, semi-simple, self-adjoint Banach algebra with unit under point-wise operations. Proof. Katznelson and McGehee have noted that B(E) is a Banach space(3). Let {f.} be a Cauchy sequence in B(E) and f its limit point in C(E). Let O + h =M(E), then IJ (f -f.) d~Ĩ IJ (f 'f~)d~+ [lf~-f~llB(~)llhAll~.
doi:10.1017/s030500410004531x fatcat:ngajr2skazgt5ftxe6iuzxjmye