Restrictions of certain function spaces to closed subgroups of locally compact groups

Abstract

Let G be a locally compact group, and E(G) either the space C U (G) of bounded left and right uniformly continuous functions on G, the space W(G) of weakly almost periodic functions on G, or the Fourier-Stieltjes algebra B(G) of G. Let E(G) \ H be the space of restrictions of i7(G)-f unctions to the closed subgroup H of G.A necessary and sufficient condition is given for an E{H)-function to belong to E(G)\ H when H is a normal subgroup of G.It is also shown that E{G)\ H is all of E(H) when H is any closed subgroup of a [SIN]-group.The techniques employed here can be used to deal with other function spaces.Let C(G) be the space of bounded continuous complex-valued functions on G with the uniform norm, || 1^, and β(G) be the Stone-Cech compactification of G, i.e., the maximal ideal space of C(G), to which C(G)-functions extend naturally, via the Gelfand transform.The left translation operator is denoted λ:[Hg)u](g')uig-'g') g, g'G, u e C(G) .

Locations

• Pacific Journal of Mathematics - View - PDF
• Project Euclid (Cornell University) - View - PDF