Extension of semicharacters and additive weights

Abstract

The central theme of this chapter is the extendability of linear multiplicative functionals from smaller to larger shift-invariant algebras. It is closely related to extendability of non-negative semicharacters and their logarithms (additive weights) from smaller to larger semigroups. We give necessary and sufficient conditions for extendability of additive weights in terms of properties such as monotonicity, and in terms of purely algebraic properties of their semigroup domains. As immediate corollaries we obtain necessary and sufficient conditions for the corresponding algebras of almost periodic functions and of H^∞-functions to possess coronae.