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Maximal Ideals in Some Spaces of Bounded Linear Operators

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

Published online by Cambridge University Press:  01 February 2018

Denny H. Leung
Denny H. Leung*
Affiliation:
Department of Mathematics, National University of Singapore, Singapore119076 (matlhh@nus.edu.sg)
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Abstract

We add to the list of Banach spaces X for which it is known that the space of bounded linear operators on X has a unique maximal ideal. In particular, the result holds if X is a subsymmetric direct sum of ℓp or of the Schlumprecht space S. We also show that two recently identified ideals in L(Jp), where Jp is the pth James space, each contains a unique maximal ideal.

Keywords

bounded linear operatorsmaximal ideal

MSC classification

Primary: 47L10: Algebras of operators on Banach spaces and other topological linear spaces 46H10: Ideals and subalgebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 251 - 264
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1. Androulakis, G. and Schlumprecht, T., The Banach space S is complementably minimal and subsequenetially prime, Studia Math. 156 (2003), 227–242.Google Scholar
2. Bird, A., Jameson, G. and Laustsen, N. J., The Giesy–James theorem for general index p, with and application to operator ideals on the pth James space, J. Operator Theory 70 (2013), 291–307.CrossRefGoogle Scholar
3. Chen, D., Johnson, W. B. and Zheng, B., Commutators on (∑ ℓ q ) p , Studia Math. 206 (2011), 175–190.Google Scholar
4. Dosev, D., On a class of operators on C(K), Houston J. Math. 41(2) (2015), 595–610.Google Scholar
5. Dosev, D. and Johnson, W. B., Commutators on ℓ∞ , Bull. Lond. Math. Soc. 42 (2010), 155–169.CrossRefGoogle Scholar
6. Dosev, D., Johnson, W. B. and Schechtman, G., Commutators on L p , 1 ≤ p < ∞, J. Amer. Math. Soc. 26 (2013), 101–127.Google Scholar
7. Kania, T. and Laustsen, N. J., Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C([0, ω1]), J. Funct. Anal. 262 (2012), 4831–4850.Google Scholar
8. Kania, T. and Laustsen, N. J., Uniqueness of the maximal ideal of operators on the ℓ p -sum of ℓ n ∞ (n ∈ ℕ) for 1 < p < ∞, Math. Proc. Camb. Phil. Soc. 160 (2016), 413–421.CrossRefGoogle Scholar
9. Laustsen, N. J., Maximal ideals in the algebra of operators on certain Banach spaces, Proc. Edinb. Math. Soc. 45 (2002), 523–546.Google Scholar
10. Laustsen, N. J., Commutators of operators on Banach spaces, J. Operator Theory 48 (2002), 503–514.Google Scholar
11. Laustsen, N. J., Loy, R. J. and Read, C. J., The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces, J. Funct. Anal. 214 (2004), 106–131.Google Scholar
12. Laustsen, N. J., Odell, E., Schlumprecht, T. and Zsák, A., Dichotomy theorems for random matrices and closed ideals of operators on (⊕∞ n = 1 ℓ n 1 ) c 0 , J. Lond. Math. Soc. 86(1) (2012), 235–258.CrossRefGoogle Scholar
13. Laustsen, N. J., Schlumprecht, T. and Zsák, A., The lattice of closed ideals in the Banach algebra of operators on a certain dual Banach space, J. Operator Theory 56 (2006), 391–402.Google Scholar
14. Leung, D. H., Ideals of operators on (⊕ ℓ∞(n))ℓ1 , Proc. Amer. Math. Soc. 143 (2015), 3047–3053.Google Scholar
15. Lin, P., Sari, B. and Zheng, B., Norm closed ideals in the algebra of bounded linear operators on Orlicz sequence spaces, Proc. Amer. Math. Soc. 142(5) (2014), 1669–1680.CrossRefGoogle Scholar
16. Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces I (Springer, 1977).Google Scholar
17. Schlumprecht, T., An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), 81–95.Google Scholar
18. Zheng, B., Commutators on (∑ ℓ p )ℓ1 , J. Math. Anal. Appl. 413(1) (2014), 284–290.Google Scholar