Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



The ideal envelope of an operator algebra

Authors: David P. Blecher and Masayoshi Kaneda
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 2103-2113
MSC (2000): Primary 46L05, 46L07, 47L30; Secondary 46H10, 47L75
Published electronically: January 27, 2004
MathSciNet review: 2053983
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A left ideal of any $C^*$-algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Conversely, we show here that operator algebras with a r.c.a.i. should be studied in terms of a certain left ideal of a $C^*$-algebra. We study operator algebras and their multiplier algebras from the perspective of ``Hamana theory'' and using the multiplier algebras introduced by the first author.

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

  • 1. D. P. Blecher, A new approach to Hilbert $C^*$-modules, Mathematische Annalen 307 (1997), 253-290. MR 98d:46063
  • 2. D. P. Blecher, The Shilov boundary of an operator space, and the characterization theorems, Journal of Functional Analysis 182 (2001), 280-343. MR 2002d:46049
  • 3. D. P. Blecher, One-sided ideals and approximate identities in operator algebras, to appear in Journal of the Australian Mathematical Society.
  • 4. D. P. Blecher and V. I. Paulsen, Multipliers of operator spaces, and the injective envelope, Pacific Journal of Mathematics 200 (2001), 1-17. MR 2002k:46150
  • 5. F. F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, New York-Heidelberg (1973). MR 54:11013
  • 6. E. G. Effros and Z. J. Ruan, Operator Spaces, The Clarendon Press, Oxford University Press, New York, 2000. MR 2002a:46082
  • 7. M. Hamana, Triple envelopes and $\Check{S}$ilov boundaries of operator spaces, Math. J. Toyama University 22 (1999), 77-93. MR 2001a:46057
  • 8. M. Kaneda and V. I. Paulsen, Characterizations of essential ideals as operator modules over $C^*$-algebras, Journal of Operator Theory 49 (2003), 245-262.
  • 9. H. Lin, Bounded module maps and pure completely positive maps, J. Operator Theory 26 (1991), 121-138. MR 94f:46071
  • 10. V. I. Paulsen, Completely bounded maps and operator algebras, Cambridge Univ. Press, 2002.
  • 11. G. Pedersen, C$^*$-algebras and their automorphism groups, Academic Press, 1979. MR 81e:46037
  • 12. G. Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, no. 294, Cambridge University Press, Cambridge, 2003.
  • 13. Y.-T. Poon and Z.-J. Ruan, Operator algebras with contractive approximate identities, Canadian Journal of Mathematics 46 (1994), 397-414. MR 95d:47057
  • 14. Z.-J. Ruan, Injectivity of operator spaces, Trans. Amer. Math. Soc. 315 (1989), 89-104. MR 91d:46078

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46L05, 46L07, 47L30, 46H10, 47L75

Retrieve articles in all journals with MSC (2000): 46L05, 46L07, 47L30, 46H10, 47L75

Additional Information

David P. Blecher
Affiliation: Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, Texas 77204-3008

Masayoshi Kaneda
Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875

Keywords: Operator algebra, nonselfadjoint algebra, ideals
Received by editor(s): November 5, 2001
Received by editor(s) in revised form: April 16, 2003
Published electronically: January 27, 2004
Additional Notes: This research was supported by a grant from the National Science Foundation
Communicated by: David R. Larson
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society