Harmonic analysis of covariant functions of characters of normal subgroups

Abstract

This paper considers L 1 L^1 -spaces of covariant functions of characters of normal subgroups. Assume that G G is a locally compact group with the group algebra L 1 ( G ) L^1(G) and N N is a closed and normal subgroup of G G . Suppose that T \mathbb {T} is the circle group and ξ : N → T \xi :N\to \mathbb {T} is a character. We introduce the Banach space L ξ 1 ( G , N ) L_\xi ^1(G,N) of covariant functions of ξ \xi on G G and present an operator theoretic approach to study structure of L ξ 1 ( G , N ) L^1_\xi (G,N) . It is proven that L ξ 1 ( G , N ) L^1_\xi (G,N) and a quotient space of L 1 ( G ) L^1(G) are isometrically isomorphic. The paper is concluded by constructive characterizations for covariant functions of ξ \xi .