On the Generalized Weighted Lebesgue Spaces of Locally Compact Groups

Let ð º be a locally compact group with a fixed left Haar measure 𠜆 and Ω be a system of weights on ð º . In this paper, we deal with locally convex space ð ¿ ð ‘ ( ð º , Ω ) equipped with the locally convex topology generated by the family of norms ( ‖ . ‖ ð ‘ , 𠜔 ) 𠜔 ∈ Ω . We study various algebraic and topological properties of the locally convex space ð ¿ ð ‘ ( ð º , Ω ) . In particular, we characterize its dual space and show that it is a semireflexive space. Finally, we give some conditions under which ð ¿ ð ‘ ( ð º , Ω ) with the convolution multiplication is a topological algebra and then characterize its closed ideals and its spectrum.