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Operator Algebras with Contractive Approximate Identities

Published online by Cambridge University Press:  20 November 2018

Yiu-Tung Poon  and
Zhong-Jin Ruan
Yiu-Tung Poon
Affiliation:
Department of Mathematics Iowa State University Ames, Iowa 50011 U.S.A.
Zhong-Jin Ruan
Affiliation:
Department of Mathematics University of Illinois, Urbana-Champaign Urbana, Illinois 61801 U.S.A.
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Abstract

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We study operator algebras with contractive approximate identities and their double centralizer algebras. These operator algebras can be characterized as L∞- Banach algebras with contractive approximate identities. We provide two examples, which show that given a non-unital operator algebra A with a contractive approximate identity, its double centralizer algebra M(A) may admit different operator algebra matrix norms, with which M(A) contains A as an M-ideal. On the other hand, we show that there is a unique operator algebra matrix norm on the unitalization algebra A1 of A such that A1 contains A as an M-ideal.

Keywords

47D2546H2547D15operator spacesoperator algebrasL∞-matricially normed spacesL∞-Banach algebrasdouble centralizer algebrasM-idealsunitalization algebras
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 2 , 01 April 1994 , pp. 397 - 414
Copyright
Copyright © Canadian Mathematical Society 1994

References

[BP] Blecher, D. and Paulsen, V., Tensor products of operator spaces, J. Funct. Anal., 99(1991), 262–292.Google Scholar
[BRS] Blecher, D., Ruan, Z-J and Sinclair, A., A characterization ofoperator algebras, J. Funct. Anal., 89(1990), 188–201.Google Scholar
[Bu] Busby, R., Double centralizers and extensions of C*-algebras, Trans. Amer. Math. Soc, 132(1968), 79- 99.Google Scholar
[CS] Christensen, E. and Sinclair, A., Representations of completely bounded multilinear operators, J. Funct. Anal., 72(1987), 151–181.Google Scholar
[Da] Davidson, K., Nest algebras, Pitman Research Notes in Mathematics Series 191, 1988.Google Scholar
[Ef] Effros, E., Advances in quantized functional analysis, Proceedings of ICM, Berkeley, (1986), 906–916.Google Scholar
[ER1] Effros, E. and Ruan, Z-J., Representations of operator bimodules and their applications, J. Operator Theory, 19(1988), 137–157.Google Scholar
[ER2] Effros, E. and Ruan, Z-J., On non-self adjoint operator algebras, Proc. Amer. Math. Soc, 110(1990), 915–922.Google Scholar
[ER3] Effros, E. and Ruan, Z-J., A new approach to operator spaces, Canad. Math. Bull. (3), 34(1991), 329–337.Google Scholar
[ER4] Effros, E. and Ruan, Z-J., Mapping spaces and liftings for operator spaces, Proc. London Math. Soc, to appear.Google Scholar
[ER5] Effros, E. and Ruan, Z-J., Operator convolution algebras: An approach to quantum groups, preprint.Google Scholar
[ER6] Effros, E. and Ruan, Z-J., On the abstract charaterization of operator spaces, Proc. Amer. Math. Soc, 119(1993), 579- 584.Google Scholar
[He] Helgason, S., Multipliers ofBanach algebras, Ann. of Math. (2), 64(1956), 240–254.Google Scholar
[Ho] Hochschlid, G., Cohomology and representations of associative algebras, Duke Math. J., 14(1947), 921–948.Google Scholar
[Jo] Johnson, B., An introduction to the theory of centralize rs, Proc. London Math. Soc. (3), 14(1964), 299–320.Google Scholar
[PR] Poon, Y. T. and Z-J. Ruan, M-ideals and quotients of subdiagonal algebras, J. Funct. Anal., 105(1992), 144–170.Google Scholar
[Rul] Ruan, Z-J., Subspacesof C*-algebras,.Funct. Anal., 76(1988), 217–230.Google Scholar
[Ru2] Ruan, Z-J., A characterization ofnon-unital operator algebras, Proc. Amer. Math. Soc, to appear.Google Scholar
[Wl] Wittstock, G., Ein operatorwertigerHahn-BanachSatz, J. Funct. Anal., 40(1981), 127–150.Google Scholar
[W2] Wittstock, G., Extensions of completely bounded C* -module homomorphism, Proc. Conference on Operator Algebras and Group Representations, Neptun (1980), Pitman, 1983.Google Scholar