Uniqueness of algebra norm on quotients of the algebra of bounded operators on a Banach space. (English) Zbl 07895009
Summary: We show that the quotient algebra \(\mathcal{B}(X) / \mathcal{I}\) has a unique algebra norm for every closed ideal \(\mathcal{I}\) of the Banach algebra \(\mathcal{B}(X)\) of bounded operators on \(X\), where \(X\) denotes any of the following Banach spaces:
- ●
- \(( \bigoplus_{n \in \mathbb{N}} \ell_2^n )_{c_0}\) or its dual space \(( \bigoplus_{n \in \mathbb{N}} \ell_2^n )_{\ell_1} \),
- ●
- \(( \bigoplus_{n \in \mathbb{N}} \ell_2^n )_{c_0} \oplus c_0(\Gamma)\) or its dual space \(( \bigoplus_{n \in \mathbb{N}} \ell_2^n )_{\ell_1} \oplus \ell_1(\Gamma)\) for an uncountable cardinal number \(\Gamma \),
- ●
- \(C_0( K_{\mathcal{A}})\), the Banach space of continuous functions vanishing at infinity on the locally compact Mrówka space \(K_{\mathcal{A}}\) induced by an uncountable, almost disjoint family \(\mathcal{A}\) of infinite subsets of \(\mathbb{N} \), constructed such that \(C_0( K_{\mathcal{A}})\) admits “few operators”.
MSC:
| 46H10 | Ideals and subalgebras |
| 46H40 | Automatic continuity |
| 47L10 | Algebras of operators on Banach spaces and other topological linear spaces |
| 46B26 | Nonseparable Banach spaces |
| 46B45 | Banach sequence spaces |
| 46E15 | Banach spaces of continuous, differentiable or analytic functions |
| 47B01 | Operators on Banach spaces |
| 47L20 | Operator ideals |
Keywords:
Banach space; Banach algebra of bounded operators; uniqueness of algebra norm; quantitative factorization of bounded operatorsReferences:
| [1] | A. Acuaviva, Factorizations and minimality of the Calkin algebra norm for \(C(K)\)-spaces, preprint, 2024. |
| [2] | Argyros, S. A.; Haydon, R. G., A hereditarily indecomposable \(\mathcal{L}_\infty \)-space that solves the scalar-plus-compact problem, Acta Math., 206, 1-54, 2011 · Zbl 1223.46007 |
| [3] | Arnott, M.; Laustsen, N. J., Classifying the closed ideals of bounded operators on two families of non-separable classical Banach spaces, J. Math. Anal. Appl., Article 125105 pp., 2021 · Zbl 1464.46056 |
| [4] | Barnes, B. A., Algebraic elements of a Banach algebra modulo an ideal, Pac. J. Math., 117, 219-231, 1985 · Zbl 0577.46048 |
| [5] | Cambern, M., On mappings of sequence spaces, Stud. Math., 30, 73-77, 1968 · Zbl 0159.41703 |
| [6] | Caradus, S. R.; Pfaffenberger, W. E.; Yood, B., Calkin Algebras and Algebras of Operators on Banach Spaces, 1974, Marcel Dekker · Zbl 0299.46062 |
| [7] | Dales, H. G., A discontinuous homomorphism from \(C(X)\), Am. J. Math., 101, 647-734, 1979 · Zbl 0417.46054 |
| [8] | Dales, H. G., Banach Algebras and Automatic Continuity, London Math. Soc. Monogr. Ser., vol. 24, 2000, Clarendon Press: Clarendon Press Oxford · Zbl 0981.46043 |
| [9] | Dales, H. G.; Woodin, W. H., An Introduction to Independence for Analysts, London Math. Soc. Lecture Note Ser., vol. 115, 1987, Cambridge University Press · Zbl 0629.03030 |
| [10] | Daws, M., Closed ideals in the Banach algebra of operators on classical non-separable spaces, Math. Proc. Camb. Philos. Soc., 140, 317-332, 2006 · Zbl 1100.46011 |
| [11] | Dowling, P. N.; Randrianantoanina, N.; Turett, B., Remarks on James’s distortion theorems II, Bull. Aust. Math. Soc., 59, 515-522, 1999 · Zbl 0947.46016 |
| [12] | Eidelheit, M., On isomorphisms of rings of linear operators, Stud. Math., 9, 97-105, 1940 · JFM 66.0548.01 |
| [13] | Esterle, J. R., Injection de semi-groupes divisibles dans des algèbres de convolution et construction d’homomorphismes discontinus de \(\mathcal{C}(K)\), Proc. Lond. Math. Soc., 36, 59-85, 1978 · Zbl 0411.46039 |
| [14] | Freeman, D.; Schlumprecht, Th.; Zsák, A., Banach spaces for which the space of operators has \(2^{\mathfrak{c}}\) closed ideals, Forum Math. Sigma, 9, 1-20, 2021 · Zbl 1467.47044 |
| [15] | Galego, E. M.; Plichko, A., On Banach spaces containing complemented and uncomplemented subspaces isomorphic to \(c_0\), Extr. Math., 18, 315-319, 2003 · Zbl 1053.46004 |
| [16] | Gohberg, I. C.; Markus, A. S.; Feldman, I. A., Normally solvable operators and ideals associated with them, Am. Math. Soc. Transl.. Am. Math. Soc. Transl., Bul. Akad. Štiince RSS Moldoven, 10, 76, 51-70, 1960, Russian original in |
| [17] | González, M.; Saksman, E.; Tylli, H.-O., Representing non-weakly compact operators, Stud. Math., 113, 265-282, 1995 · Zbl 0832.47039 |
| [18] | Gramsch, B., Eine Idealstruktur Banachscher Operatoralgebren, J. Reine Angew. Math., 225, 97-115, 1967 · Zbl 0145.16602 |
| [19] | Hagler, J.; Johnson, W. B., On Banach spaces whose dual balls are not weak* sequentially compact, Isr. J. Math., 28, 325-330, 1977 · Zbl 0365.46019 |
| [20] | Horváth, B.; Kania, T., Surjective homomorphisms from algebras of operators on long sequence spaces are automatically injective, Quart. J. Math., 72, 1167-1189, 2021 · Zbl 1526.46033 |
| [21] | James, R. C., Uniformly non-square Banach spaces, Ann. Math., 80, 542-550, 1964 · Zbl 0132.08902 |
| [22] | Johnson, B. E., Continuity of homomorphisms of algebras of operators, J. Lond. Math. Soc., 42, 537-541, 1967 · Zbl 0152.32805 |
| [23] | Johnson, W. B.; Kania, T.; Schechtman, G., Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support, Proc. Am. Math. Soc., 144, 4471-4485, 2016 · Zbl 1372.46017 |
| [24] | W.B. Johnson, N.C. Phillips, Manuscript in preparation. |
| [25] | Kania, T.; Kochanek, T., The ideal of weakly compactly generated operators acting on a Banach space, J. Oper. Theory, 71, 455-477, 2014 · Zbl 1313.47177 |
| [26] | Kania, T.; Laustsen, N. J., Ideal structure of the algebra of bounded operators acting on a Banach space, Indiana Univ. Math. J., 66, 1019-1043, 2017 · Zbl 1422.47068 |
| [27] | Koszmider, P., On decompositions of Banach spaces of continuous functions on Mrówka’s spaces, Proc. Am. Math. Soc., 133, 2137-2146, 2005 · Zbl 1085.46015 |
| [28] | Koszmider, P.; Laustsen, N. J., A Banach space induced by an almost disjoint family, admitting only few operators and decompositions, Adv. Math., 381, Article 107613 pp., 2021 · Zbl 07319242 |
| [29] | Laustsen, N. J.; Loy, R. J.; Read, C. J., The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces, J. Funct. Anal., 214, 106-131, 2004 · Zbl 1067.46026 |
| [30] | Laustsen, N. J.; Schlumprecht, Th.; Zsák, A., The lattice of closed ideals in the Banach algebra of operators on a certain dual Banach space, J. Oper. Theory, 56, 391-402, 2006 · Zbl 1104.47062 |
| [31] | Laustsen, N. J.; Skillicorn, R., Extensions and the weak Calkin algebra of Read’s Banach space admitting discontinuous derivations, Stud. Math., 236, 51-62, 2017 · Zbl 1371.46038 |
| [32] | Lindenstrauss, J.; Tzafriri, L., Classical Banach Spaces, Vol. I, Ergeb. Math. Grenzgeb., vol. 92, 1977, Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0362.46013 |
| [33] | Luft, E., The two-sided closed ideals of the algebra of bounded linear operators of a Hilbert space, Czechoslov. Math. J., 18, 595-605, 1968 · Zbl 0197.11301 |
| [34] | Meyer, M. J., On a topological property of certain Calkin algebras, Bull. Lond. Math. Soc., 24, 591-598, 1992 · Zbl 0785.46046 |
| [35] | Meyer, M. J., Lower bounds for norms on certain algebras, Ill. J. Math., 39, 567-575, 1995 · Zbl 0845.46025 |
| [36] | Motakis, P., Separable spaces of continuous functions as Calkin algebras, J. Am. Math. Soc., 2023 |
| [37] | Motakis, P.; Puglisi, D.; Zisimopoulou, D., A hierarchy of separable commutative Calkin algebras, Indiana Univ. Math. J., 65, 39-67, 2016 · Zbl 1353.46014 |
| [38] | Namioka, I.; Phelps, R. R., Banach spaces which are Asplund spaces, Duke Math. J., 42, 735-750, 1975 · Zbl 0332.46013 |
| [39] | Palmer, T. W., Banach Algebras and the General Theory of^⁎-Algebras, Vol. I, 1994, Cambridge University Press · Zbl 0809.46052 |
| [40] | Pełczyński, A., On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular operators in \(C(S)\)-spaces, Bull. Acad. Pol. Sci., 13, 31-37, 1965 · Zbl 0138.38604 |
| [41] | Pietsch, A., Operator Ideals, 1980, North Holland · Zbl 0399.47039 |
| [42] | Read, C. J., Discontinuous derivations on the algebra of bounded operators on a Banach space, J. Lond. Math. Soc., 40, 305-326, 1989 · Zbl 0722.46020 |
| [43] | Rosenthal, H. P., On relatively disjoint families of measures, with some applications to Banach space theory, Stud. Math., 37, 13-36, 1970 · Zbl 0227.46027 |
| [44] | Rudin, W., Continuous functions on compact spaces without perfect subsets, Proc. Am. Math. Soc., 8, 39-42, 1957 · Zbl 0077.31103 |
| [45] | Tarbard, M., Hereditarily indecomposable, separable \(\mathcal{L}_\infty\) Banach spaces with \(\ell_1\) dual having few but not very few operators, J. Lond. Math. Soc., 85, 737-764, 2012 · Zbl 1257.46013 |
| [46] | Tylli, H.-O., Exotic algebra norms on quotient algebras of the bounded linear operators, (Filali, M.; Arhippainen, J., Proceedings of the \(3^{\text{rd}}\) International Conference on Topological Algebra and Applications, Oulu 2-6 July, 2001, 2004, Oulu University Press: Oulu University Press Oulu), 210-224 · Zbl 1076.47060 |
| [47] | Ware, G. K., Uniqueness of norm properties of Calkin algebras, 2014, Australian National University, available online at |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
