Isometric uniqueness of a complementably universal Banach space for Schauder decompositions. (English) Zbl 1286.46012

A Banach space \(U\) is called complementably universal for a given class of Banach spaces provided that \(U\) is a member of this class and any space of this class is isomorphic to a complemented subspace of \(U\). The first example of a complementably universal space for the class of spaces with a basis was given by A. Pełczyński [Stud. Math. 32, 247–268 (1969; Zbl 0185.37401)]. This example was followed by an example of a space complementably universal for the class of Banach spaces with the bounded approximation property by M. I. Kadets [Stud. Math. 40, 85–98 (1971; Zbl 0218.46015)] and a complementably universal space for the class of spaces with a finite-dimensional decomposition (FDD) by A. Pełczyński and P. Wojtaszczyk [Stud. Math. 40, 91–108 (1971; Zbl 0221.46014)]. All these spaces are isomorphic by the Pełczyński decomposition method.
In the paper, an isometric version of the above spaces is presented, i.e., a Banach space \(\mathbb{P}\) such that any Banach space with a monotone FDD is isometric to a 1-complemented subspace of \(\mathbb{P}\). Again, by the Pełczyński decomposition method, the space \(\mathbb{P}\) is isomorphic to the aforementioned universal spaces. The author also proves isometric homogeneity of \(\mathbb{P}\) (i.e., that any linear isometry between any two 1-complemented finite-dimensional subspaces of \(\mathbb{P}\) extends to a bijective isometry of \(\mathbb{P}\)) and isometric uniqueness of \(\mathbb{P}\) among spaces satisfying a certain extension-type property. The construction is based on the category theory technique extensively used in the study of universal and “homogeneous” objects for different types of classes, as the space constructed by P. Urysohn [Bull. Math. (2) 51, 43–64 (1927); 51, 74–90 (1927; JFM 53.0556.01)] within the class of separable metric spaces and the space of V. I. Gurarij [Sib. Mat. Zh. 7, 1002–1013 (1966; Zbl 0166.39303)] in the class of separable Banach spaces, see, for example, [W. Kubiś and S. Solecki, Isr. J. Math. 195, Part A, 449–456 (2013; Zbl 1290.46010)].


46B04 Isometric theory of Banach spaces
46M15 Categories, functors in functional analysis
46M40 Inductive and projective limits in functional analysis
Full Text: DOI arXiv Euclid


[1] A. Avilés, F. Cabello Sánchez, J.M.F. Castillo, M. González and Y. Moreno, Banach spaces of universal disposition , J. Funct. Anal. 261 (2011), 2347-2361. · Zbl 1236.46014
[2] M. Droste and R. Göbel, A categorical theorem on universal objects and its application in abelian group theory and computer science , Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989), 49-74, Contemp. Math., 131, Part 3, Amer. Math. Soc., Providence, RI, 1992. · Zbl 0759.18002
[3] M. Fabian, P. Habala, P. Hájek, V. Montesinos and V. Zizler, Banach Space Theory. The Basis for Linear and Nonlinear Analysis , CMS Books in Mathematics. Springer, New York, 2011. · Zbl 1229.46001
[4] R. Fraïssé, Sur quelques classifications des systèmes de relations , Publ. Sci. Univ. Alger. Sér. A. 1 (1954), 35-182. · Zbl 0068.24302
[5] W.B. Johnson and A. Szankowski, Complementably universal Banach spaces , Studia Math. 58 (1976), 91-97. · Zbl 0341.46017
[6] M. I. Kadec, On complementably universal Banach spaces , Studia Math. 40 (1971), 85-89. · Zbl 0218.46015
[7] W. Kubiś, Fraïssé sequences: category-theoretic approch to universal homogeneus structures , preprint, arxiv.org/abs/0711.1683.
[8] W. Kubiś and S. Solecki, A proof of uniqueness of the Gurarii space , to appear in Israel J. Math., arxiv.org/abs/1110.0903. · Zbl 1290.46010
[9] A. Pełczyński, Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basis , Studia Math. 40 (1971), 239-243. · Zbl 0223.46019
[10] A. Pełczyński, Universal bases , Studia Math. 32 (1969), 247-268. · Zbl 0185.37401
[11] A. Pełczyński, Projections in certain Banach spaces , Studia Math. 19 (1960), 209-228. · Zbl 0104.08503
[12] A. Pełczyński and P. Wojtaszczyk, Banach spaces with finite dimensional expansions of identity and universal bases of finite dimensional subspaces , Studia Math. 40 (1971), 91-108. · Zbl 0221.46014
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.