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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.
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