Strongly zero product determined Banach algebras. (English) Zbl 1543.47116
The paper under review is concerned with the study of certain classes of Banach algebras from the point of view of being zero product determined.
More precisely, the following quantitative version of the property “zero product determined” is investigated. For the Banach algebra \(A\), there exists a constant \(\alpha\) such that, for every continuous bilinear functional \(\varphi: A \times A \to \mathbb{C}\), there exists a continuous linear functional \(\xi\) on \(A\) such that \[\sup_{\|a\| =1 = \|b\|} |\varphi(a,b)- \xi(a,b) | \leqslant \alpha \sup_{\|a\| =1 = \|b\|, \; ab=0} | \varphi(a,b)| \] in each of the following cases:
In each of the above cases, the precise value of \(\alpha\) is given.
More precisely, the following quantitative version of the property “zero product determined” is investigated. For the Banach algebra \(A\), there exists a constant \(\alpha\) such that, for every continuous bilinear functional \(\varphi: A \times A \to \mathbb{C}\), there exists a continuous linear functional \(\xi\) on \(A\) such that \[\sup_{\|a\| =1 = \|b\|} |\varphi(a,b)- \xi(a,b) | \leqslant \alpha \sup_{\|a\| =1 = \|b\|, \; ab=0} | \varphi(a,b)| \] in each of the following cases:
- (i)
- \(A\) is a \(C^*\)-algebra;
- (ii)
- \(A = L^1(G)\), where \(G\) is a locally compact group;
- (iii)
- \(A = \mathcal{A}(X)\), the algebra of approximable operators on a Banach space \(X\) which has property \(X\).
In each of the above cases, the precise value of \(\alpha\) is given.
Reviewer: Bence Horváth (Praha)
MSC:
| 47H60 | Multilinear and polynomial operators |
| 42A20 | Convergence and absolute convergence of Fourier and trigonometric series |
| 47L10 | Algebras of operators on Banach spaces and other topological linear spaces |
References:
| [1] | Alaminos, J.; Brešar, M.; Extremera, J.; Villena, A. R., Maps preserving zero products, Stud. Math., 193, 131-159 (2009) · Zbl 1168.47029 |
| [2] | Alaminos, J.; Brešar, M.; Extremera, J.; Villena, A. R., Zero Lie product determined Banach algebras, Stud. Math., 239, 189-199 (2017) · Zbl 1394.46036 |
| [3] | Alaminos, J.; Brešar, M.; Extremera, J.; Villena, A. R., Zero Lie product determined Banach algebras, II, J. Math. Anal. Appl., 474, 2, 1498-1511 (2019) · Zbl 1426.46030 |
| [4] | Alaminos, J.; Brešar, M.; Extremera, J.; Villena, A. R., Zero Jordan product determined Banach algebras, J. Aust. Math. Soc., 1-14 (2020) |
| [5] | Alaminos, J.; Extremera, J.; Godoy, M. L.C.; Villena, A. R., Hyperreflexivity of the space of module homomorphisms between non-commutative \(L^p\)-spaces, J. Math. Anal. Appl., 498, 2, Article 124964 pp. (2021) · Zbl 1519.46045 |
| [6] | Alaminos, J.; Extremera, J.; Villena, A. R., Approximately zero product preserving maps, Isr. J. Math., 178, 1-28 (2010) · Zbl 1209.47013 |
| [7] | Alaminos, J.; Extremera, J.; Villena, A. R., Hyperreflexivity of the derivation space of some group algebras, Math. Z., 266, 571-582 (2010) · Zbl 1203.47019 |
| [8] | Alaminos, J.; Extremera, J.; Villena, A. R., Hyperreflexivity of the derivation space of some group algebras, II, Bull. Lond. Math. Soc., 44, 323-335 (2012) · Zbl 1239.47026 |
| [9] | Allan, G. R.; Ransford, T. J., Power-dominated elements in a Banach algebra, Stud. Math., 94, 1, 63-79 (1989) · Zbl 0705.46021 |
| [10] | Brešar, M., Zero Product Determined Algebras, Frontiers in Mathematics (2021), Birkhäuser: Birkhäuser Basel · Zbl 1505.16001 |
| [11] | Dales, H. G., Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, New Series, vol. 24 (2000), Oxford Science Publications, the Clarendon Press, Oxford University Press: Oxford Science Publications, the Clarendon Press, Oxford University Press New York · Zbl 0981.46043 |
| [12] | Runde, V., Amenable Banach algebras. A Panorama, Springer Monographs in Mathematics (2020), Springer-Verlag: Springer-Verlag New York · Zbl 1445.46001 |
| [13] | Samei, E., Reflexivity and hyperreflexivity of bounded n-cocycles from group algebras, Proc. Am. Math. Soc., 139, 163-176 (2011) · Zbl 1222.47056 |
| [14] | Samei, E.; Soltani Farsani, J., Hyperreflexivity of the bounded n-cocycle spaces of Banach algebras, Monatshefte Math., 175, 429-455 (2014) · Zbl 1317.47037 |
| [15] | Samei, E.; Soltani Farsani, J., Hyperreflexivity constants of the bounded n-cocycle spaces of group algebras and \(C^\ast \)-algebras, J. Aust. Math. Soc., 109, 112-130 (2020) · Zbl 1456.46045 |
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.
