Abstract
The Gurariy space G is defined by the property that for every pair of finite dimensional Banach spaces L ⊂ M, every isometry T: L → G admits an extension to an isomorphism \(\mathop T\limits^ \sim :M \to G\) with ‖T‖‖T −1‖ ≤ 1 + ∈. We investigate the question when we can take \(\mathop T\limits^ \sim \) to be also an isometry (i.e., ∈ = 0). We identify a natural class of pairs L ⊂ M such that the above property for this class with ∈ = 0 characterises the Gurariy space among all separable Banach spaces. We also show that the Gurariy space G is the only Lindenstrauss space such that its finite-dimensional smooth subspaces are dense in all subspaces.
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References
A. Aviles, F. Cabello, J. Castillo, M. Gonzalez and Y. Morena, Banach spaces of universal disposition, Journal of Functional Analysis 261 (2011), 2347–2361.
V. I. Gurariy, Space of universal disposition, isotopic spaces and the Mazur problem on rotation of Banach spaces, Sibirskiĭ Matematicheskiĭ Zhurnal 7 (1966), 1002–1013.
J. Garbulińska and W. Kubiś, Remarks on Gurarii spaces, Extracta Mathematicae 26 (2011), 235–269.
W. Kubiś and S. Solecki, A proof of uniqueness of Gurarii space, Israel Journal of Mathematics 195 (2013), 449–456.
A. J. Lazar and J. Lindenstrauss, Banach spaces whose duals are L 1 spaces and their representing matrices, Acta Mathematica 126 (1971), 165–193.
W. Lusky, The Gurariy spaces are unique, Archiv der Mathematik 27 (1976), 627–635.
W. Lusky, On separable Lindenstrauss spaces, Journal of Functional Analysis 26 (1977), 103–120.
P. Wojtaszczyk, Some remarks on the Gurariy space, Studia Mathematica 41 (1972), 207–2101.
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We dedicate this paper to the memory of Joram Lindenstrauss, a fine man and a great mathematician, whose ideas influenced us through our entire professional lives.
Both authors were partially supported by the Foundation for Polish Science. The first-named author was partially supported by the Israel Science Foundation, Grant 209/09 and by European Grant Spade 2. The second-named author was partially supported by the Center for Advanced Studies in Mathematics of Ben-Gurion University of the Negev, the “HPC Infrastructure for Grand Challenges of Science and Engineering Project, co-financed by the European Regional Development Fund under the Innovative Economy Operational Programme and Polish NCN grant DEC2011/03/B/ST1/04902.
The authors would like to express their gratitude to the anonymous referee for his/her penetrating criticism which lead to a substantial revision of the paper.
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Fonf, V.P., Wojtaszczyk, P. Characteristic properties of the Gurariy space. Isr. J. Math. 203, 109–140 (2014). https://doi.org/10.1007/s11856-014-0016-4
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DOI: https://doi.org/10.1007/s11856-014-0016-4