Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-16T17:23:51.643Z Has data issue: false hasContentIssue false
  • English
  • Français

Representation of a quotient of a subalgebra of B(X)

Published online by Cambridge University Press:  24 October 2008

Christian Le Merdy
Christian Le Merdy
Affiliation:
Equipe de Mathématiques, UA CNRS 741, Université de Franche-Comté, 25030 Besançon Cedex, France
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X be an SQp-space, i.e. a quotient of a subspace of some Lp-space. Let B ⊂ B(X) be a subalgebra of all bounded operators on X and let I ⊂ B be a closed ideal. We show that the quotient algebra B/I is isometrically homomorphic to a subalgebra of B(Y) for some SQp-space Y. This generalizes a theorem of Bernard and Cole, corresponding to p = 2, which states that any quotient of an operator algebra is an operator algebra.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 1 , January 1996 , pp. 83 - 90
Copyright
Copyright © Cambridge Philosophical Society 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bernard, A.. Quotients of operator algebras. Seminar on uniform algebras. University of Aberdeen, 1973.Google Scholar
[2]Blecher, D. P.. Tensor product of operator spaces II. Canadian J. Math. 44 (1992), 75–90.CrossRefGoogle Scholar
[3]Blecher, D. P.. A completely bounded characterization of operator algebras. To appear in Math. Annalen.Google Scholar
[4]Blecher, D. P. and Paulsen, V. I.. Tensor product of operator spaces. J. Funct. Anal. 99 (1991), 262–292.Google Scholar
[5]Blecher, D. P.. Ruan, Z.-J. and Sinclair, A. M.. A characterization of operator algebras. J. Funet. Anal. 89 (1990), 188–201.Google Scholar
[6]Bonsall, F. F. and Duncan, J.. Complete normed algebras (Springer-Verlag, 1973).CrossRefGoogle Scholar
[7]Christensen, E. and Sinclair, A. M.. Representations of completely bounded multilinear operators. J. Funct. Anal. 72 (1987), 151–181.Google Scholar
[8]Davie, A. M.. Quotient algebras of uniform algebras. J. London Math. Soc. 7 (1973), 31–40.Google Scholar
[9]Dixon, P. G.. Varieties of Banach algebras. Quart. J. Math. Oxford 27 (1976), 481–487.CrossRefGoogle Scholar
[10]Effros, E. G. and Ruan, Z.-J.. A new approach to operator spaces. Canadian Math. Bull. 34 (1991), 329–337.CrossRefGoogle Scholar
[11]Effros, E. G. and Ruan, Z.-J.. On the abstract characterization of operator spaces. Proc. Amer. Math. Soc. 119 (1993), 579–584.CrossRefGoogle Scholar
[12]Hernandez, H.. Espaces Lp, factorisation et produits tensoriels dans les espaces de Banach. C.R. Acad. Sci. Paris 296 (1983), 385–388.Google Scholar
[13]Kwapien, S.. On operators factorizable through Lp-spaces. Bull. Soc. Math. France Mem. 31–32 (1972), 215–225.Google Scholar
[14]Le Merdy, C.. Factorization of p–completely bounded multlinear maps. Pacific J. of Math, (to appear).Google Scholar
[15]Lumer, G.. Etats, algèbres quotients et sous-espaces invariants. C.R. Acad. Sci. Paris 274 (1972), 1308–1311.Google Scholar
[16]Paulsen, V. I. and Smith, R. R.. Multilinear maps and tensor norms on operator systems. J. Fund. Anal. 73 (1987), 258–276.Google Scholar
[17]Pisier, G.. Completely bounded maps between sets of Banach space operators. Indiana Univ. Math. J. 39 (1990), 249–277.CrossRefGoogle Scholar
[18]Pisier, G.. Factorization of linear operators and geometry of Banach spaces. CBMS Vol. 60 (American Math. Soc., 1987).Google Scholar
[19]Ruan, Z.-J.. Subspaces of C*-algebras. J. Funct. Anal. 76 (1988), 217–230.CrossRefGoogle Scholar
[20]Varopoulos, N. T.. A theorem on operator algebras. Math. Scand. 37 (1975), 173–182.Google Scholar
[21]Varopoulos, N. T.. Some remarks on Q-algebras. Ann. Inst. Fourier 22 (1972), 1–11.CrossRefGoogle Scholar
[22]Werner, J.. Quotient algebras of uniform algebras. Symposium on function algebras and rational approximation. University of Michigan, 1969.Google Scholar