Banach spaces in various positions. (English) Zbl 1208.46007
Recall that J.Lindenstrauss and H.P.Rosenthal in [“Automorphisms in \(c_{0}\), \(l_{1}\) and \(m\)”, Isr.J.Math.7, 227–239 (1969; Zbl 0186.18602)] showed that \(c_{0}\) has the property that every isomorphism between two of its infinite codimensional subspaces can be extend to an automorphism of the whole space and formulated the so-called automorphic space problem. Are \(c_{0}\) and \(l_{2}\) the only separable Banach spaces with that property?
The paper under review outgrows from the study of different aspects of that problem as the authors described in the excellent summary: “We formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index \(\mathfrak a(Y, X)\) that measures how many non-equivalent positions \(Y\) admits in \(X\), and obtain estimates of \(\mathfrak a(Y, X)\) for \(X\) a classical Banach space such as \(\ell _p, L_p, L_{1},C(\omega ^{\omega })\) or \(C[0,1]\). Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from \(c_{0}\) or \(\ell _{2}\)? Recall that a Banach space \(X\) is said to be automorphic if every subspace \(Y\) admits only one position in \(X\); i.e., \(\mathfrak a(Y, X)=1\) for every subspace \(Y\) of \(X\). We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an \(\mathcal L_{\infty }\)-space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both \(X\) and \(X^{*}\) are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either \(c_{0}\) or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either \(L_{\infty }\) or a superreflexive type 2 Banach lattice.”
The paper under review outgrows from the study of different aspects of that problem as the authors described in the excellent summary: “We formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index \(\mathfrak a(Y, X)\) that measures how many non-equivalent positions \(Y\) admits in \(X\), and obtain estimates of \(\mathfrak a(Y, X)\) for \(X\) a classical Banach space such as \(\ell _p, L_p, L_{1},C(\omega ^{\omega })\) or \(C[0,1]\). Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from \(c_{0}\) or \(\ell _{2}\)? Recall that a Banach space \(X\) is said to be automorphic if every subspace \(Y\) admits only one position in \(X\); i.e., \(\mathfrak a(Y, X)=1\) for every subspace \(Y\) of \(X\). We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an \(\mathcal L_{\infty }\)-space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both \(X\) and \(X^{*}\) are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either \(c_{0}\) or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either \(L_{\infty }\) or a superreflexive type 2 Banach lattice.”
Reviewer: Elói M. Galego (Sao Paulo)
MSC:
| 46B03 | Isomorphic theory (including renorming) of Banach spaces |
| 46B25 | Classical Banach spaces in the general theory |
Citations:
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