Generalized group algebras and generalized measure algebrason non-discrete locally compact abelian groups

LetGbe a non-discrete LCA group with the dual group Gamma. We definea generalized group algebra,L1(G), and a generalized measure algebra,M(G), onGasgeneralizations of the group algebraL1(G) and the measure algebraM(G), respectively.Generalized Fourier transforms of elements ofL1(G) and generalized Fourier-Stieltjestransforms of elements ofM(G) are also defined as generalizations of the Fourier trans-forms and the Fourier-Stieltjes transforms, respectively. The imageA(Gamma) ofL1(G) bythe generalized Fourier transform becomes a function algebra on Gamma with norm inher-ited fromL1(G) through this transform. It is shown thatA(Gamma) is a natural Banachfunction algebra on Gamma which is BSE and BED. It turns out thatL1(G) contains all Ra-jchman measures. Segal algebras inL1(G) are defined and investigated. It is shown thatthere exists the smallest isometrically character-invariant Segal algebra inL1(G), which(eventually) coincides with the smallest isometrically character-invariant Segal algebrainL1(G), the Feichtinger algebra ofG. A notion of locally bounded elements ofM(G)is introduced and investigated. It is shown that for each locally bounded element mu ofM(G) there corresponds a unique Radon measure iota mu onGwhich characterizes mu.We investigate the multiplier algebraM(L1(G)) ofL1(G), and obtain a result thatthere is a natural continuous isomorphism fromM(L1(G)) intoA(G)& lowast;, the algebra ofpseudomeasures onG. WhenGis compact, this map becomes surjective and isometric