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Let T be a unitary operator on
a Hilbert space H. Then in particular,
(i) T is a contraction, i.e. ∥T∥≦ 1; and
(ii) The spectrum of T is a subset of the unit circle, i.e. Sp (T) ⊂ C, where C
denotes the set of complex numbers of absolute value one.
Call an arbitrary operator T a unimodular contraction if it satisfies conditions (i)
and (ii) above. Then several questions immediately come to mind. Do there
exist nonunitary unimodular contractions? If so, what is the nature of their
spectra, e.g. what subsets of the unit circle arise as spectra of nonunitary
unimodular contractions; when does the spectrum contain point, residual, or
continuous spectrum? Under what conditions is a unimodular contraction unitary?
What is the nature of operator algebras containing nonunitary unimodular
contractions?
In this paper examples are given of nonunitary unimodular contractions. It is shown
(Theorem 2) that such exist with arbitrarily prescribed spectrum, which however can
contain no residual spectrum. It is also shown (Theorem 1) that nonunitary
unimodular contractions exist only in infinite von Neumann algebras. This result is
applied to a mapping problem of operator algebras.
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