The Dichotomy Problem for Homogeneous Banach Algebras

Abstract

Let A(T) denote the algebra of absolutely convergent Fourier series, and let C(T) denote the algebra of continuous functions on the circle group T. A classical result of Katznelson asserts that only analytic functions operate on A(T). On the other hand, all continuous functions operate on C(T). In this note, we construct a Banach algebra which is natural in the sense of harmonic analysis, but which enjoys a far different symbolic calculus. Specifically, we show the existence of a strongly homogeneous Banach algebra B so that (1) A(T) !; B ; C(T) and (2) non-analytic functions operate on B. Our results provide a negative solution to the dichotomy problem in the context of homogeneous Banach algebras (see [4] and [5]). The construction, as well as some notations and further comments, will appear in the succeeding sections. 1. Let B be a commutative, semi-simple, self-adjoint Banach algebra with maximal ideal space T. We view B as an algebra of continuous functions on T. B will be called homogeneous provided the following two properties hold: ( 1 ) For every a e T, the mapping f(x) -4 f(x + a) is an isometry of B into itself. ( 2 ) For every f e B, we have lima0 I | (x + a) f(x) lB 0. B will be called strongly homogeneous provided we also have ( 3 ) For every integer k, the operator f(x) -f(kx) maps B into itself and is of norm 1. Let B1 and B, be homogeneous Banach algebras. A function F: [-1, 1]->C is said to operate from B1 into B, provided the following obtains: For every f e B, with -1 ?f ? 1, we have Fof e B2. In case B1 = B2, we say simply that F operates on B1. We will prove the following theorem. THEOREM 1. There exists a strongly homogeneous Banach algebra B