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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
DOI: https://doi.org/10.1090/S0002-9939-1970-0259622-4
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.


References [Enhancements On Off] (What's this?)

  • [1] John F. Aarnes and Richard V. Kadison, Pure states and approximate identities, Proc. Amer. Math. Soc. 21 (1969), 749-752. MR 0240633 (39:1980)
  • [2] A. Brown, P. R. Halmos and A. L. Shields, Cesàro operators, Acta Sci. Math. Szeged 26 (1965), 125-137. MR 32 #4539. MR 0187085 (32:4539)
  • [3] J. Dixmier, Les $ {C^{\ast}}$-algèbres et leurs représentations, Cahiers Scientifiques, fasc. 29, Gauthier-Villars, Paris, 1964. MR 30 #1404. MR 0171173 (30:1404)
  • [4] R. G. Douglas, On majorization, factorization and range inclusion of [ill]perators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413-415. MR 34 #3315. MR 0203464 (34:3315)
  • [5] L. Terrell Gardner, On isomorphisms of $ {C^{\ast}}$-algebras, Amer. J. Math. 87 (1965), 384-396. MR 31 #3883. MR 0179637 (31:3883)
  • [6] Paul R. Halmos, Introduction to Hilbert space, Chelsea, New York, 1957.
  • [7] -, A Hilbert space problem book, Van Nostrand, Princeton, N. J., 1967. MR 34 #8178. MR 0208368 (34:8178)
  • [8] R. V. Kadison and I. Singer, Extensions of pure states, Amer. J. Math. 81 (1959), 383-400. MR 23 #A1243. MR 0123922 (23:A1243)
  • [9] Bela Sz.-Nagy and Ciprian Foiaş, Analyse harmonique des opérateurs de l'espace de Hilbert, Masson, Paris and Akadémiai Kiadó, Budapest, 1967. MR 37 #778.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0259622-4
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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