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Some more twisted Hilbert spaces. (English) Zbl 1476.46016

A twisted Hilbert space is a Banach space \(X\) that admits a Hilbert subspace \(H\) so that \(X/H\) is also a Hilbert space. As follows from the definition, they share with Hilbert spaces all 3-space properties of Hilbert spaces (to be reflexive, to be \(\ell_2\)-saturated, etc.)and for other properties they are, more often than not, close to be Hilbert (they have type \(p\) for all \(p<2\) and cotype \(q\) for all \(q>2\), etc.). The question is therefore if (and how) it is possible to construct twisted Hilbert spaces that are either very different from a Hilbert spaces or very similar, without being Hilbert. A survey-like paper on the topic is [J. M. F. Castillo, Adv. Anal. Geom. 2, 43–66 (2020; Zbl 1476.46073)]. Twisted Hilbert spaces are not a pathology in modern Banach space theory: they constitute, for various reasons, essential and rather ubiquitous objects.
The top five list of papers on twisted Hilbert spaces is: first, [P. Enflo et al., Math. Scand. 36, 199–210 (1975; Zbl 0314.46015)], where it is shown that non-Hilbert twisted Hilbert spaces exist; and second [N. J. Kalton and N. T. Peck, Trans. Am. Math. Soc. 255, 1–30 (1979; Zbl 0424.46004)], where it is shown how to construct them and that twisted Hilbert spaces are intimately connected with complex interpolation, a topic that Kalton establishes and develops in [N. J. Kalton, Trans. Am. Math. Soc. 333, No. 2, 479–529 (1992; Zbl 0776.46033)]. Third, [N. J. Kalton, J. Inst. Math. Jussieu 2, No. 3, 401–408 (2003; Zbl 1035.46007)], where it is shown that a twisted Hilbert space with unconditional basis is a Hilbert space. Fourth is the paper [J. Suárez de la Fuente, J. Inst. Math. Jussieu 19, No. 3, 855–867 (2020; Zbl 1447.46015)] where the author fulfils the seemingly impossible task of constructing a twisted Hilbert weak Hilbert space that is not Hilbert.
The present paper could well be the fifth: following the lead of the fourth one, the authors obtain three new examples of very singular twisted Hilbert spaces: an asymptotically Hilbertian but not weak Hilbert space, a non-asymptotically Hilbertian space and a HAPpy space (a space all whose subspaces have the Approximation Property), and many others.

MSC:

46B06 Asymptotic theory of Banach spaces
46B70 Interpolation between normed linear spaces
46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
46B45 Banach sequence spaces
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References:

[1] Albiac, F., and N. J. Kalton: Topics in Banach space theory. -Grad. Texts in Math. 233, Springer-Verlag. · Zbl 1188.46013
[2] Alon, N., H. Kaplan, G. Nivasch, M. Sharir, and S. Smorodinsky: Weak ǫ-nets and interval chains. -In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, San Francisco, California, USA, January 20-22, 2008, 1194-1203. · Zbl 1192.68812
[3] Bergh, J., and J. Löfström: Interpolation spaces. An introduction. -Grundlehren Math. Wiss. 223. Springer-Verlag, Berlin-New York, 1976. · Zbl 0344.46071
[4] Calderón, A. P.: Intermediate spaces and interpolation, the complex method. -Studia Math. 24, 1964, 113-190. · Zbl 0204.13703
[5] Carro, M. J., J. Cerdà, and J. Soria: Commutators and interpolation methods. -Ark. Mat. 33:2, 1995, 199-216. · Zbl 0903.46069
[6] Casazza, P. G., and N. J. Nielsen: A Banach space with a symmetric basis which is of weak cotype 2 but not of cotype 2. -Studia Math. 157:1, 2003, 1-16. · Zbl 1032.46011
[7] Casazza, P., and T. J. Shura: Tsirelson’s space. Lecture Notes in Math. 1363, Springer-Verlag, Berlin, 1989. · Zbl 0709.46008
[8] Castillo, J. M. F., D. Morales, and J. Suárez de la Fuente: Derivation of vector-valued complex interpolation scales. -J. Math. Anal. Appl. 468:1, 2018, 461-472. · Zbl 1479.46085
[9] Cobos, F., and T. Schonbek: On a theorem by Lions and Peetre about interpolation between a Banach space and its dual. -Houston J. Math. 24:2, 1998, 325-344. · Zbl 1161.46309
[10] Enflo, P., J. Lindenstrauss, and G. Pisier: On the “three space problem”. -Math. Scand. 36:2, 1975, 199-210. · Zbl 0314.46015
[11] Ferenczi, V., and C. Rosendal: Ergodic Banach spaces. -Adv. Math. 195, 2005, 259-282. · Zbl 1082.46009
[12] Johnson, W. B.: Banach spaces all of whose subspaces have the approximation property. -In: Seminar on Functional Analysis, 1979-1980 (French), Exp. No. 16, École Polytech., Palaiseau, 1980. 1-11. · Zbl 0441.46014
[13] Johnson, W. B., and A. Naor: The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite. -Discrete Comput. Geom. 43:3, 2010, 542-553. · Zbl 1196.46013
[14] Johnson, W. B., and A. Szankowski: Hereditary approximation property. -Ann. of Math. (2) 176:3, 2012, 1987-2001. · Zbl 1272.46012
[15] Kalton, N. J.: Differentials of complex interpolation processes for Köthe function spaces. -Trans. Amer. Math. Soc. 333:2, 1992, 479-529. · Zbl 0776.46033
[16] Kalton, N. J.: Twisted Hilbert spaces and unconditional structure. -J. Inst. Math. Jussieu 2:3, 2003, 401-408. · Zbl 1035.46007
[17] Kalton, N. J., and S. Montgomery-Smith: Interpolation of Banach spaces. -Handbook of Geometry of Banach Spaces, Vol. 2, Elsevier, Amsterdam, 2003, 1131-1175. · Zbl 1041.46012
[18] Kalton, N. J., and N. T. Peck: Twisted sums of sequence spaces and the three-space prob-lem. -Trans. Amer. Math. Soc. 255, 1979, 1-30. · Zbl 0424.46004
[19] Kwapień, S.: Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. -Studia Math. 44, 1972, 583-595. · Zbl 0256.46024
[20] Maurey, B.: Un théorème de prolongement. -C.R. Acad. Sci. Paris A279, 1974, 329-332. · Zbl 0291.47001
[21] Maurey, B., V. D. Milman, and N. Tomczak-Jaegermann: Asymptotic infinite-dimensional theory of Banach spaces. -In: Geometric aspects of functional analysis (Israel, 1992-1994), 149-175. · Zbl 0872.46013
[22] Milman, V. D., and G. Schechtman: Asymptotic theory of finite dimensional normed spaces. -Lecture Notes in Math. 1200, Springer-Verlag, Berlin, 1986. · Zbl 0606.46013
[23] Nielsen, N. J., and N. Tomczak-Jaegermann: Banach lattices with property (H) and weak Hilbert spaces. -Illinois J. Math. 36:3 1992, 345-371. · Zbl 0787.46018
[24] Pisier, P.: Weak Hilbert spaces. -Proc. London Math. Soc. (3) 56:3, 1988, 547-579. · Zbl 0666.46009
[25] Pisier, G.: Volume of convex bodies and Banach space geometry. -Cambridge Tracts in Math. 94. Cambridge Univ. Press, Cambridge, 1989. · Zbl 0698.46008
[26] Pisier, G.: Factorization of linear operators and geometry of Banach spaces. -CBMS Regional Conference Series in Mathematics 60, Amer. Math. Soc., Providence, RI, 1986). · Zbl 0588.46010
[27] Suárez de la Fuente, J.: A weak Hilbert space that is a twisted Hilbert space. -J. Inst. Math. Jussieu 19:3, 2020, 855-867. · Zbl 1447.46015
[28] Suárez de la Fuente, J.: A space with no unconditional basis that satisfies the Johnson-Lindenstrauss lemma. -Results Math. 74:3, 2019, Art. 126, 1-14 · Zbl 1429.46008
[29] Suárez de la Fuente, J.: A universal formula for derivation maps with applications. -Submitted.
[30] Suárez de la Fuente, J.: The Kalton-Peck space as a spreading model. -Submitted.
[31] Tomczak-Jaegermann, N.: Computing 2-summing norms with few vectors. -Ark. Mat. 17:2, 1979, 273-277. · Zbl 0436.47033
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