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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Characters on singly generated $ C\sp{\ast} $-algebras

Author: John Bunce
Journal: Proc. Amer. Math. Soc. 25 (1970), 297-303
MSC: Primary 46.65; Secondary 47.00
MathSciNet review: 0259622
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Abstract: In this note we consider the question of what elements $ \delta $ in the spectrum of a bounded operator $ A$ on Hilbert space have the property that there is a multiplicative linear functional $ \phi $ on the $ {C^{\ast}}$-algebra generated by $ A$ and $ I$ whose value at $ A$ is $ \delta $. If $ A$ is hyponormal then there is a character $ \phi $ on the $ {C^{\ast}}$-algebra generated by $ A$ and $ I$ such that $ \phi (A) = \delta $ if and only if $ \delta $ is in the approximate point spectrum of $ A$. We use this to prove a structure theorem for the $ {C^{\ast}}$-algebra generated by a hyponormal operator. We conclude by proving that any pure state on a Type I $ {C^{\ast}}$-algebra is multiplicative on some maximal abelian $ {C^{\ast}}$-subalgebra.

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Keywords: Character, approximate point spectrum, irreducible representation, universal representation, pure state, atomic representation, Radon-Nikodým theorem, hyponormal operator, Cesàro operator, irreducible operator, weighted shift, maximal abelian subalgebras
Article copyright: © Copyright 1970 American Mathematical Society

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