Universal decomposed Banach spaces. (English) Zbl 1457.46012

Banach spaces that are universal in some way are always interesting objects. For instance, \(C[0,1]\) is separable and contains isometric copies of all separable Banach spaces. The Gurariy space \(\mathcal G\) [V. I. Gurarij, Sib. Mat. Zh. 7, 1002–1013 (1966; Zbl 0166.39303)] has the property that, given two finite-dimensional Banach spaces \(E\subset F\), every into isometry \(E\to \mathcal G\) extends, for every \(\varepsilon>0\), to an \(\varepsilon\)-isometry \(F\to \mathcal G\). A Banach space having the former property is said to be universal for the class of separable spaces; having the latter, to be of almost universal disposition.
Let us now add the word “complemented”. W. B. Johnson and A. Szankowski showed in [Stud. Math. 58, 91–97 (1976; Zbl 0341.46017)] that no complemented version of \(C[0,1]\) (i.e., a separable Banach space containing complemented copies of all separable Banach spaces) is possible. However, M. I. Kadets constructed in [Stud. Math. 40, 85–98 (1971; Zbl 0218.46015)] the complemented + BAP version \(\mathcal K\): a separable Banach space with the Bounded Approximation Property (BAP) containing complemented copies of all separable Banach spaces with the BAP. J. Garbulińska-Węgrzyn showed in [Banach J. Math. Anal. 8, No. 1, 211–220 (2014; Zbl 1286.46012)] that \(\mathcal K\) can be constructed as a Fraïssé limit and noticed several noteworthy properties of \(\mathcal K\). J. M. F. Castillo and Y. Moreno showed in [Q. J. Math. 71, No. 1, 139–174 (2020; Zbl 1447.46007)] that the multiple pushout technique developed by A. Avilés et al. [J. Funct. Anal. 261, No. 9, 2347–2361 (2011; Zbl 1236.46014)] to construct \(\mathcal G\) and new versions of \(\mathcal G\) could be used to construct \(\mathcal K\) and new versions of \(\mathcal K\) and to prove that the mentioned noteworthy properties of Garbulińska-Węgrzyn [loc. cit.] can be refined to show that \(\mathcal K\) is the “complemented” version of \(\mathcal G\): a space of universal “complemented” disposition. The paper of Cabello, Castillo and Moreno [F. Cabello et al., Forum Math. 31, No. 6, 1533–1556 (2019; Zbl 1444.46015)] focuses instead on the Fraïssé techniques to construct and study the \(p\)-Banach versions of \(\mathcal K\). The \(p\)-Banach versions of \(\mathcal G\) were obtained, via the multiple pushout technique, by F. Cabello Sánchez et al. [J. Funct. Anal. 267, No. 3, 744–771 (2014; Zbl 1321.46003)].
A careful analysis and detailed exposition of all that will be found in the forthcoming book [F. Cabello and J. M. F. Castillo, Homological methods in Banach spaces, Cambridge Studies in Advanced Mathematics, Cambridge Univ. Press, Chapters 6 and 8].
In the paper under review, the authors return to the Fraïssé approach to construct new universal spaces for the newly introduced class of \(\mathfrak B\)-decomposed Banach spaces, which are Banach spaces admitting a certain decomposition in terms of the spaces of a prefixed class \(\mathfrak B\) of finite-dimensional spaces.


46B04 Isometric theory of Banach spaces
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46M15 Categories, functors in functional analysis
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