Characterization of homogeneous spaces and their norms

Abstract

Let G denote a compact group and B a homogeneous Banach space of pseudomeasures over G {Bis left translation invariant with continuous shifts). If T(G) is the linear space of trigonometric polynomials defined on G then T(G) B is a dense subspace of B. An explicit description is given of T(G) B. A complete list of the homogeneous closed subspaces of B is given. Moreover it is shown that B is determined by the set T(G) B and N 9 the restriction of the B norm to this set. This leads to a complete description of those subsets of T(G) and norms N which determine homogeneous Banach spaces.