The ideal envelope of an operator algebra

Blecher, David; Kaneda, Masayoshi

doi:10.1090/S0002-9939-04-07303-4

The ideal envelope of an operator algebra
HTML articles powered by AMS MathViewer

by David P. Blecher and Masayoshi Kaneda PDF

Proc. Amer. Math. Soc. 132 (2004), 2103-2113 Request permission

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

David P. Blecher, A new approach to Hilbert $C^*$-modules, Math. Ann. 307 (1997), no. 2, 253–290. MR 1428873, DOI 10.1007/s002080050033
David P. Blecher, The Shilov boundary of an operator space and the characterization theorems, J. Funct. Anal. 182 (2001), no. 2, 280–343. MR 1828796, DOI 10.1006/jfan.2000.3734
D. P. Blecher, One-sided ideals and approximate identities in operator algebras, to appear in Journal of the Australian Mathematical Society.
David P. Blecher and Vern I. Paulsen, Multipliers of operator spaces, and the injective envelope, Pacific J. Math. 200 (2001), no. 1, 1–17. MR 1863404, DOI 10.2140/pjm.2001.200.1
Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029, DOI 10.1007/978-3-642-65669-9
Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753
Masamichi Hamana, Triple envelopes and Šilov boundaries of operator spaces, Math. J. Toyama Univ. 22 (1999), 77–93. MR 1744498
M. Kaneda and V. I. Paulsen, Characterizations of essential ideals as operator modules over $C^*$-algebras, Journal of Operator Theory 49 (2003), 245-262.
Hua Xin Lin, Bounded module maps and pure completely positive maps, J. Operator Theory 26 (1991), no. 1, 121–138. MR 1214924
V. I. Paulsen, Completely bounded maps and operator algebras, Cambridge Univ. Press, 2002.
Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
G. Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, no. 294, Cambridge University Press, Cambridge, 2003.
Yiu Tung Poon and Zhong-Jin Ruan, Operator algebras with contractive approximate identities, Canad. J. Math. 46 (1994), no. 2, 397–414. MR 1271223, DOI 10.4153/CJM-1994-021-0
Zhong-Jin Ruan, Injectivity of operator spaces, Trans. Amer. Math. Soc. 315 (1989), no. 1, 89–104. MR 929239, DOI 10.1090/S0002-9947-1989-0929239-3