Ideals of operators on (⊕ℓ^{∞}(𝑛))_{ℓ¹}

Leung, Denny

doi:10.1090/S0002-9939-2015-12500-2

Ideals of operators on $(\oplus \ell ^\infty (n))_{\ell ^1}$

Author: Denny H. Leung
Journal: Proc. Amer. Math. Soc. 143 (2015), 3047-3053
MSC (2010): Primary 47L10; Secondary 46H10
DOI: https://doi.org/10.1090/S0002-9939-2015-12500-2
Published electronically: February 20, 2015
MathSciNet review: 3336629
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The unique maximal ideal in the Banach algebra $L(E)$, $E = (\oplus \ell ^\infty (n))_{\ell ^1}$, is identified. The proof relies on techniques developed by Laustsen, Loy and Read (published in the Journal of Functional Analysis in 2004) and a dichotomy result for operators mapping into $L^1$ due to Laustsen, Odell, Schlumprecht and Zsák (published in the Journal of the London Mathematical Society in 2012).

References

Spiros A. Argyros and Richard G. Haydon, A hereditarily indecomposable $\scr L_\infty $-space that solves the scalar-plus-compact problem, Acta Math. 206 (2011), no. 1, 1–54. MR 2784662, DOI https://doi.org/10.1007/s11511-011-0058-y
J. W. Calkin, Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math. (2) 42 (1941), 839–873. MR 5790, DOI https://doi.org/10.2307/1968771

I.C. Gohberg, A.S. Markus, I.A. Feldman, Normally solvable operators and ideals associated with them, AMS Translation 61(1967), 63-84.

Bernhard Gramsch, Eine Idealstruktur Banachscher Operatoralgebren, J. Reine Angew. Math. 225 (1967), 97–115 (German). MR 206727, DOI https://doi.org/10.1515/crll.1967.225.97

Anna Kamińska, Alexey I. Popov, Eugeniu Spinu, Adi Tcaciuc, and Vladimir G. Troitsky, Norm closed operator ideals in Lorentz sequence spaces, J. Math. Anal. Appl. 389 (2012), no. 1, 247–260. MR 2876495, DOI https://doi.org/10.1016/j.jmaa.2011.11.034

Tomasz Kania and Niels Jakob Laustsen, Uniqueness of the maximal ideal of the Banach algebra of bounded operators on $C([0,\omega _1])$, J. Funct. Anal. 262 (2012), no. 11, 4831–4850. MR 2913688, DOI https://doi.org/10.1016/j.jfa.2012.03.011

Niels Jakob Laustsen, Richard J. Loy, and Charles J. Read, The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces, J. Funct. Anal. 214 (2004), no. 1, 106–131. MR 2079887, DOI https://doi.org/10.1016/j.jfa.2004.02.009

N. J. Laustsen, E. Odell, Th. Schlumprecht, and A. Zsák, Dichotomy theorems for random matrices and closed ideals of operators on $(\bigoplus ^\infty _{n=1}\ell ^n_1)c_0$, J. Lond. Math. Soc. (2) 86 (2012), no. 1, 235–258. MR 2959303, DOI https://doi.org/10.1112/jlms/jdr083

Niels Jakob Laustsen, Thomas Schlumprecht, and András Zsák, The lattice of closed ideals in the Banach algebra of operators on a certain dual Banach space, J. Operator Theory 56 (2006), no. 2, 391–402. MR 2282689

Peikee Lin, Bünyamin Sari, and Bentuo Zheng, Norm closed ideals in the algebra of bounded linear operators on Orlicz sequence spaces, Proc. Amer. Math. Soc. 142 (2014), no. 5, 1669–1680. MR 3168473, DOI https://doi.org/10.1090/S0002-9939-2014-11903-4

Erhard Luft, The two-sided closed ideals of the algebra of bounded linear operators of a Hilbert space, Czechoslovak Math. J. 18(93) (1968), 595–605. MR 238086

B. Sari, Th. Schlumprecht, N. Tomczak-Jaegermann, and V. G. Troitsky, On norm closed ideals in $L(l_p,l_q)$, Studia Math. 179 (2007), no. 3, 239–262. MR 2291733, DOI https://doi.org/10.4064/sm179-3-3

Th. Schlumprecht, On the closed subideals of $L(\ell _p\oplus \ell _q)$, Oper. Matrices 6 (2012), no. 2, 311–326. MR 2976049, DOI https://doi.org/10.7153/oam-06-22