An extremely non-homogeneous weak Hilbert space (Q2841377)

A Banach space \(X\) is called a weak Hilbert space provided that there are \(C,\delta>0\) such that any finite-dimensional subspace \(E\) of \(X\) contains a further subspace \(F\) with the following properties: \(\dim F\geq \delta \dim E\), \(F\) is \(C\)-complemented in \(X\), and the Banach-Mazur distance between \(F\) and \(\ell_2^{\dim F}\) is at most \(C\). The notion originated in papers by \textit{V. D.~Milman} and \textit{G.~Pisier} [Isr. J. Math. 54, 139--158 (1986; Zbl 0611.46022)] and \textit{G.~Pisier} [Proc. Lond. Math. Soc., III. Ser. 56, No. 3, 547--579 (1988; Zbl 0666.46009)]. The class of weak Hilbert spaces shares many regular properties of Hilbert spaces, in particular, it is closed under taking subspaces, quotients and duals, and every weak Hilbert space is superreflexive. However, there are examples of weak Hilbert spaces not containing \(\ell_2\); the canonical example is the convexified Tsirelson space [\textit{W. B.~Johnson}, Stud. Math. 55, 201--205 (1976; Zbl 0362.46015)]. A natural question concerns the regularity of the global structure of weak Hilbert spaces.NEWLINENEWLINEThe authors present an example of a weak Hilbert space with extremely non-homogeneous global structure. The constructed Banach space \(X_{wh}\) with an unconditional basis shares the following properties: (1) \(X_{wh}\) is strongly asymptotic \(\ell_2\), which ensures that \(X_{wh}\) is weak Hilbert. (2) For any block subspace \(Y\) of \(X_{wh}\), any bounded operator on \(Y\) is a strictly singular perturbation of a restriction of a diagonal operator on \(X_{wh}\), in particular: (3) None of the block subspaces of \(X_{wh}\) is isomorphic to one of its proper subspaces. (4) No disjointly supported subspaces of \(X_{wh}\) are isomorphic, i.e., \(X_{wh}\) is tight by support.NEWLINENEWLINEThe technical construction of the space \(X_{wh}\) uses the framework of spaces with non-regular properties built on the basis of mixed Tsirelson spaces. The underlying modified mixed Tsirelson space, defined with the help of Schreier families, ensures the strong asymptotic \(\ell_2\) structure of \(X_{wh}\), in a similar manner to the case of the hereditarily indecomposable strongly asymptotic \(\ell_1\) space of \textit{S.A.~Argyros, I.~Deliyanni, D.~N.~Kutzarova} and \textit{A.~Manoussakis} [J. Funct. Anal. 159, No. 1, 43--109 (1998; Zbl 0931.46017)]. The strong non-homogeneous structure of \(X_{wh}\), described in properties (2)--(4), is guaranteed by the use of a coding function. Coding functions, introduced by \textit{B.~Maurey} and \textit{H.~P.~Rosenthal} [Stud. Math. 61, 77--98 (1977; Zbl 0357.46025)] in the construction of a weakly null sequence with no unconditional subsequence, became the crucial tool in building spaces with few operators or few symmetries. In the unconditional setting, coding functions were first used by \textit{W. T.~Gowers} [Bull. Lond. Math. Soc. 26, No. 6, 523--530 (1994; Zbl 0838.46011)] in his solution of the Banach hyperplane problem.NEWLINENEWLINEThe authors recall an open question of P.\,Casazza (cf.\ [\textit{G. Androulakis} et al., Can. Math. Bull. 43, No.~3, 257--267 (2000; Zbl 0977.46007)]), whether there exists a hereditarily indecomposable weak Hilbert space.