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One-sided ideals and approximate identities in operator algebras

Part of: Linear spaces and algebras of operators Selfadjoint operator algebras Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  09 April 2009

David P. Blecher
David P. Blecher
Affiliation:
Department of Mathematics, University of Houston, 4800 Calhoun, Houston, TX 77204-3008, USA e-mail: [email protected]
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Abstract

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A left ideal on any C*-algebra is an example of an operator algebra with a right contractive approximate indentiy (r.c.a.i.). Indeed, left ideal in C*-algebras may be charcterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper, that operator algebras with r.c.a.i. shoulod be studied in terms of a certain let ideal of a C*-algebra. We study left ideals from the perspective of ‘Hamana theory’ and using the multiplier algebras of an operator space studied elsewhere by the author. More generally, we develop some general theory for operator algebras which have a 1-sided identity or approzimate indentity, including a Banach-Stone theorem for these algebras, and an analysis of the ‘multiplier operator algebra’.

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras 46L07: Operator spaces and completely bounded maps 47L30: Abstract operator algebras on Hilbert spaces 46H10: Ideals and subalgebras 47L75: Other nonselfadjoint operator algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 3 , June 2004 , pp. 425 - 448
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Arveson, W. B., ‘Subalgebras of C*-algebras’, Acta Math. 123 (1969), 141–224.CrossRefGoogle Scholar
[2]Arveson, W. B., ‘Subalgebras of C*-algebras II’, Acta Math. 128 (1972), 271–308.CrossRefGoogle Scholar
[3]Blecher, D. P., ‘Commutativity in operator algebras’, Proc. Amer. Math. Soc. 109 (1990), 709–715.CrossRefGoogle Scholar
[4]Blecher, D. P., ‘A generalization of Hilbert modules’, J. Funct. Anal. 136 (1996), 365–421.CrossRefGoogle Scholar
[5]Blecher, D. P., ‘Multipliers and dual operator algebras’, J. Funct. Anal. 183 (2001), 498–525.CrossRefGoogle Scholar
[6]Blecher, D. P., ‘The Shilov boundary of an operator space, and the characterization theorems’, J. Funct. Anal. 182 (2001), 280–343.CrossRefGoogle Scholar
[7]Blecher, D. P., Effros, E. G. and Zarikian, V., ‘One-sided M-ideals and multipliers in operator spaces I’, Pacific J. Math. 206 (2002), 287–319.CrossRefGoogle Scholar
[8]Blecher, D. P. and Kaneda, M., ‘The ideal envelope of an operator algebra’, Proc. Amer. Math. Soc. (to appear).Google Scholar
[9]Blecher, D. P., Muhly, P. S. and Paulsen, V. I., Categories of operator modules—Morita equivalence and projective modules, Mem. Amer. Math. Soc. 143 (2000), no. 681 (Amer. Math. Soc., Providence, 2000).Google Scholar
[10]Blecher, D. P. and Paulsen, V. I., ‘Multipliers of operator spaces, and the injective envelope’, Pacific J. Math. 200 (2001), 1–17.CrossRefGoogle Scholar
[11]Blecher, D. P., Ruan, Z. J. and Sinclair, A.M., ‘A characterization of operator algebras’, J. Funct. Anal. 89 (1990), 188–201.CrossRefGoogle Scholar
[12]Blecher, D. P. and Solel, B., ‘A double commutant theorem for operator algebras’, J. Operator Theory (to appear).Google Scholar
[13]Bonsall, F. F. and Duncan, J., Complete normed algebras (Springer, New York, 1973).CrossRefGoogle Scholar
[14]Conway, J. B., A course in functional analysis (Springer, New York, 1985).CrossRefGoogle Scholar
[15]Effros, E. G., Ozawa, N. and Ruan, Z. J., ‘On injectivity and nuclearity for operator spaces’, Duke Math. J. 110 (2001), 489–521.CrossRefGoogle Scholar
[16]Effros, E. G. and Ruan, Z. J., ‘On non-self-adjoint operator algebras’, Proc. Amer. Math. Soc. 110 (1990), 915–922.Google Scholar
[17]Effros, E. G. and Ruan, Z. J., Operator spaces (Oxford Univ. Press, Oxford, 2000).Google Scholar
[18]Amin, K. El, Campoy, A. M. and Rodriguez-Palacios, A., ‘A holomorphic characterization of C*-and JB*-algebras’, Manuscripta Math. 104 (2001), 467–478.CrossRefGoogle Scholar
[19]Hamana, M., ‘Triple envelopes and Silov boundaries of operator spaces’, Math. J. Toyama Univ. 22 (1999), 77–93.Google Scholar
[20]Kadison, R. V., ‘Isometries of operator algebras’, Ann. of Math. 54 (1951), 325–338.CrossRefGoogle Scholar
[21]Merdy, C. Le, ‘An operator space characterization of dual operator algebras’, Amer. J. Math., 121 (1999), 55–63.CrossRefGoogle Scholar
[22]Neal, M. and Russo, B., ‘Operator space characterizations of C*-algebras. and ternary rings’, Pacific J. Math. 209 (2003), 339–364.CrossRefGoogle Scholar
[23]Palmer, T. W., Banach algebras and the general theory of *-algebras. Vol. 1, Encyclopedia of mathematics and its applications Vol. 49 (Cambridge Univ. Press, Cambridge, 1994).CrossRefGoogle Scholar
[24]Paulsen, V. I., ‘Unpublished notes on multiplier algebras’, circa 1992.Google Scholar
[25]Paulsen, V. I., Completely bounded maps and operator algebras (Cambridge Univ. Press, Cambridge, 2002).Google Scholar
[26]Pederson, G., C*-algebras and their automorphism groups (Academic Press, 1979).Google Scholar
[27]Pisier, G., Introduction to operator space theory (Cambridge Univ. Press, Cambridge, 2003).CrossRefGoogle Scholar
[28]Poon, Y.-T. and Ruan, Z. J., ‘Operator algebras with contractive approximate identities’, Canad. J. Math. 46 (1994), 397–414.CrossRefGoogle Scholar
[29]Wegge-Olsen, N. E., K-theory and C*-algebras (Oxford Univ. Press, Oxford, 1993).CrossRefGoogle Scholar
[30]Werner, W., ‘Small K-groups for operator systems’, preprint, 1999.Google Scholar
[31]Zettl, H., ‘A Characterization of ternary rings of operators’, Adv. Math. 48 (1983), 117–143.CrossRefGoogle Scholar
[32]Blecher, D. P. and Merdy, C. Le, Operator algebras and their modules (Oxford Univ. Press, Oxford, to appear).CrossRefGoogle Scholar
[33]Blecher, D. P. and Zarikian, V., ‘The calculus on one-sided M-ideal and multipliers in operator spaces’, preprint, 2003.Google Scholar
[34]Willis, G. A., ‘Factorization in finite-dimensional ideals of group algebras’, Proc. London Math. Soc. 82 (2001), 676–700.CrossRefGoogle Scholar