Shift-invariant algebras on compact groups

Abstract

In this chapter we introduce shift-invariant algebras, the main objects of this book. These are uniform algebras on a compact connected group G, consisting of continuous functions on G, whose spectrum is contained in a semigroup S of the dual group Ĝ. If Ĝ is a subgroup of ℝ, and S ⊂ ℝ+, then the maximal ideal space of the corresponding shift-invariant algebra is the G-disc, or, big disc over G. In this chapter we describe two important models of shift-invariant algebras, namely, by the means of almost periodic functions on ℝ, and by the means of H∞-functions on the unit circle. The set of automorphisms, and the peak groups of shift-invariant algebras are also characterized. Extensions on G-discs and groups of several classical theorems of Complex Analysis, such as Radó’s theorem for null-sets and the Riemann theorem for removable singularities of analytic functions, are stated and proved. It is shown that these extensions hold for some semigroups, while in general they fail. In principle we state all results for general shift-invariant algebras AS, though they apply automatically to the particular cases of algebras APS of almost periodic functions, and of H S ∞ -algebras. Asymptotically almost periodic functions combine the properties of classical almost periodic functions on ℝ, and of continuous functions on ℝ that vanish at infinity.