Vol. 21, No. 1, 1967

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Certain isomorphisms between quotients of a group algebra

O. Carruth McGehee

Vol. 21 (1967), No. 1, 133–152

Let T be the circle group, considered as the additive group of the real numbers modulo 2π. Let A = A(T), the Banach algebra of functions on T which have absolutely convergent Fourier series, with the norm of f in A equal to n|f(n)|. If E is a closed subset of T, we denote by A(E) the quotient algebra A∕I(E), where I(E) is the closed ideal consisting of those functions in A which vanish on E. This paper presents a procedure for constructing perfect sets E and F, which are not Helson sets, and a map φ : F E inducing an isomorphism of A(E) into A(F). Thereby we shall obtain cases of an isomorphism of norm one, where φ is the restriction to F of a discontinuous character of T, composed with a rotation. In general, our φ will be such a restriction at least on a dense subset of F, with the norm of the isomorphism not necessarily equal to one.

In the course of this construction we impose a condition of “arithmetic thinness” on the set F. As we shall prove, this condition is sufficient to imply that F is a set of uniqueness.

Mathematical Subject Classification
Primary: 42.56
Received: 19 May 1966
Published: 1 April 1967
O. Carruth McGehee