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

Garbulińska-Węgrzyn, Joanna

Banach J. Math. Anal. 8, No. 1, 211-220 (2014).

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)].

Reviewer: Anna Pelczar-Barwacz (Kraków)

MSC:

46B04	Isometric theory of Banach spaces
46M15	Categories, functors in functional analysis
46M40	Inductive and projective limits in functional analysis

Keywords:

complementably universal Banach space; finite-dimensional decomposition; Fraïssé sequence

Citations:

Zbl 0185.37401; Zbl 0218.46015; Zbl 0221.46014; Zbl 0166.39303; JFM 53.0556.01; Zbl 1290.46010

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