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Compact elements and smallest faithful representation of $ C\sp *$-algebras


Authors: Raphael A. Hauser and Heinrich Matzinger
Journal: Proc. Amer. Math. Soc. 123 (1995), 3379-3384
MSC: Primary 46L05; Secondary 47B10
DOI: https://doi.org/10.1090/S0002-9939-1995-1285995-7
MathSciNet review: 1285995
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Abstract: Let A be a $ {C^ \ast }$-algebra which either allows a faithful separable representation or is postliminal. We prove that A then admits a smallest faithful representation if and only if the ideal of compact elements is an essential ideal in A.


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

  • [1] B. A. Barnes, G. J. Murphy, M. R. Smyth, and T. T. West, Riesz and Fredholm theory in Banach algebras, Research Notes in Math., vol. 67, Pitman, London, 1982.
  • [2] C. Constantinescu, Funktionalanalysis, Autographie zur Vorlesung an der ETH Zürich, 1992.
  • [3] J. Dixmier, Von Neumann algebras, North-Holland, Amsterdam, 1981. MR 641217 (83a:46004)
  • [4] -, $ {C^ \ast }$-algebras, North-Holland, Amsterdam, 1982.
  • [5] J. A. Erdos, On certain elements of $ {C^ \ast }$-algebras, Illinois J. Math. 15 (1971), 682-693. MR 0290120 (44:7305)
  • [6] J. M. G. Fell and R. S. Doran, Representations of $ \ast $-algebras, locally compact groups, and Banach $ \ast $-algebraic bundles, Vol. 1, Academic Press, New York, 1988.
  • [7] G. J. Murphy, $ {C^ \ast }$-algebras and operator theory, Academic Press, San Diego, CA, 1990. MR 1074574 (91m:46084)
  • [8] M. A. Naimark, On a problem of the theory of rings with involution, Uspekhi Mat. Nauk (N.S.) 6 (1951), 160-164. (Russian) MR 0046574 (13:755f)
  • [9] G. K. Pedersen, $ {C^ \ast }$-algebras and their automorphism groups, London Math. Soc. Monograph, no. 14, Academic Press, New York, 1979. MR 548006 (81e:46037)
  • [10] A. Rosenberg, The number of irreducible representations of simple rings with no minimal ideals, Amer. J. Math. 75 (1953), 523-530. MR 0057477 (15:236f)
  • [11] K. Vala, Sur les elements compacts d'une algèbre normée, Ann. Acad. Sci. Fenn. 407 (1967). MR 0222642 (36:5692)
  • [12] K. Ylinen, Compact and finite-dimensional elements of normed algebras, Ann. Acad. Sci. Fenn. 428 (1968). MR 0238089 (38:6365)
  • [13] -, A note on the compact elements of $ {C^ \ast }$-algebras, Proc. Amer. Math. Soc. 35 (1972), 305-306. MR 0296716 (45:5775)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1285995-7
Article copyright: © Copyright 1995 American Mathematical Society

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