Operator Algebras with Contractive Approximate Identities

Poon, Yiu-Tung ; Ruan, Zhong-Jin

Canadian journal of mathematics, Tome 46 (1994) no. 2, pp. 397-414

Résumé

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 A 1 of A such that A 1 contains A as an M-ideal.

Détail

DOI : 10.4153/CJM-1994-021-0

Mots-clés : 47D25, 46H25, 47D15, operator spaces, operator algebras, L ∞-matricially normed spaces, L ∞-Banach algebras, double centralizer algebras, M-ideals, unitalization algebras

Bibliographie

[BP] [BP] Blecher, D. and Paulsen, V., Tensor products of operator spaces, J. Funct. Anal., 99(1991), 262–292. Google Scholar

[BRS] [BRS] Blecher, D., Ruan, Z-J and Sinclair, A., A characterization ofoperator algebras, J. Funct. Anal., 89(1990), 188–201. Google Scholar

[Bu] [Bu] Busby, R., Double centralizers and extensions of C*-algebras, Trans. Amer. Math. Soc, 132(1968), 79- 99. Google Scholar

[CS] [CS] Christensen, E. and Sinclair, A., Representations of completely bounded multilinear operators, J. Funct. Anal., 72(1987), 151–181. Google Scholar

[Da] [Da] Davidson, K., Nest algebras, Pitman Research Notes in Mathematics Series 191, 1988. Google Scholar

[Ef] [Ef] Effros, E., Advances in quantized functional analysis, Proceedings of ICM, Berkeley, (1986), 906–916. Google Scholar

[ER1] [ER1] Effros, E. and Ruan, Z-J., Representations of operator bimodules and their applications, J. Operator Theory, 19(1988), 137–157. Google Scholar

[ER2] [ER2] Effros, E. and Ruan, Z-J., On non-self adjoint operator algebras, Proc. Amer. Math. Soc, 110(1990), 915–922. Google Scholar

[ER3] [ER3] Effros, E. and Ruan, Z-J., A new approach to operator spaces, Canad. Math. Bull. (3), 34(1991), 329–337. Google Scholar

[ER4] [ER4] Effros, E. and Ruan, Z-J., Mapping spaces and liftings for operator spaces, Proc. London Math. Soc, to appear. Google Scholar

[ER5] [ER5] Effros, E. and Ruan, Z-J., Operator convolution algebras: An approach to quantum groups, preprint. Google Scholar

[ER6] [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] [He] Helgason, S., Multipliers ofBanach algebras, Ann. of Math. (2), 64(1956), 240–254. Google Scholar

[Ho] [Ho] Hochschlid, G., Cohomology and representations of associative algebras, Duke Math. J., 14(1947), 921–948. Google Scholar

[Jo] [Jo] Johnson, B., An introduction to the theory of centralize rs, Proc. London Math. Soc. (3), 14(1964), 299–320. Google Scholar

[PR] [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] [Rul] Ruan, Z-J., Subspacesof C*-algebras,.Funct. Anal., 76(1988), 217–230. Google Scholar

[Ru2] [Ru2] Ruan, Z-J., A characterization ofnon-unital operator algebras, Proc. Amer. Math. Soc, to appear. Google Scholar

[Wl] [Wl] Wittstock, G., Ein operatorwertigerHahn-BanachSatz, J. Funct. Anal., 40(1981), 127–150. Google Scholar

[W2] [W2] Wittstock, G., Extensions of completely bounded C* -module homomorphism, Proc. Conference on Operator Algebras and Group Representations, Neptun (1980), Pitman, 1983. Google Scholar