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
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