Twisted sums with 𝐶(𝐾) spaces

Sánchez, F.; Castillo, J.; Kalton, N.; Yost, D.

doi:10.1090/S0002-9947-03-03152-0

Twisted sums with $C(K)$ spaces
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by F. Cabello Sánchez, J. M. F. Castillo, N. J. Kalton and D. T. Yost

Trans. Amer. Math. Soc. 355 (2003), 4523-4541

DOI: https://doi.org/10.1090/S0002-9947-03-03152-0

Published electronically: July 2, 2003

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Abstract:

If $X$ is a separable Banach space, we consider the existence of non-trivial twisted sums $0\to C(K)\to Y\to X\to 0$, where $K=[0,1]$ or $\omega ^{\omega }.$ For the case $K=[0,1]$ we show that there exists a twisted sum whose quotient map is strictly singular if and only if $X$ contains no copy of $\ell _1$. If $K=\omega ^{\omega }$ we prove an analogue of a theorem of Johnson and Zippin (for $K=[0,1]$) by showing that all such twisted sums are trivial if $X$ is the dual of a space with summable Szlenk index (e.g., $X$ could be Tsirelson’s space); a converse is established under the assumption that $X$ has an unconditional finite-dimensional decomposition. We also give conditions for the existence of a twisted sum with $C(\omega ^{\omega })$ with strictly singular quotient map.

References

Israel Aharoni and Joram Lindenstrauss, Uniform equivalence between Banach spaces, Bull. Amer. Math. Soc. 84 (1978), no. 2, 281–283. MR 482074, DOI 10.1090/S0002-9904-1978-14475-9
D. Amir, Continuous functions’ spaces with the separable projection property, Bull. Res. Council Israel Sect. F 10F (1962), 163–164 (1962). MR 150570
D. Amir, Projections onto continuous function spaces, Proc. Amer. Math. Soc. 15 (1964), 396–402. MR 165350, DOI 10.1090/S0002-9939-1964-0165350-3
John Warren Baker, Projection constants for $C(S)$ spaces with the separable projection property, Proc. Amer. Math. Soc. 41 (1973), 201–204. MR 320707, DOI 10.1090/S0002-9939-1973-0320707-8
Y. Benyamini and J. Lindenstrauss, A predual of $l_{1}$ which is not isomorphic to a $C(K)$ space, Israel J. Math. 13 (1972), 246–254 (1973). MR 331013, DOI 10.1007/BF02762798
Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
Aleksander Błaszczyk and Andrzej Szymański, Concerning Parovičenko’s theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), no. 7-8, 311–314 (1981) (English, with Russian summary). MR 628044
J. Bourgain and F. Delbaen, A class of special ${\cal L}_{\infty }$ spaces, Acta Math. 145 (1980), no. 3-4, 155–176. MR 590288, DOI 10.1007/BF02414188
Jean Bourgain and Gilles Pisier, A construction of ${\scr L}_{\infty }$-spaces and related Banach spaces, Bol. Soc. Brasil. Mat. 14 (1983), no. 2, 109–123. MR 756904, DOI 10.1007/BF02584862
Y. A. Brudnyi and N. J. Kalton, Polynomial approximation on convex subsets of $\textbf {R}^n$, Constr. Approx. 16 (2000), no. 2, 161–199. MR 1735240, DOI 10.1007/s003659910008
Félix Cabello Sánchez and Jesús M. F. Castillo, Duality and twisted sums of Banach spaces, J. Funct. Anal. 175 (2000), no. 1, 1–16. MR 1774849, DOI 10.1006/jfan.2000.3598
F. Cabello Sánchez and J. M. F. Castillo, Uniform boundedness and twisted sums of Banach spaces, Houston J. Math., to appear.
Félix Cabello Sánchez, Jesús M. F. Castillo, and David Yost, Sobczyk’s theorems from A to B, Extracta Math. 15 (2000), no. 2, 391–420. III Congress on Banach Spaces (Jarandilla de la Vera, 1998). MR 1823898
Peter G. Casazza and Thaddeus J. Shura, Tsirel′son’s space, Lecture Notes in Mathematics, vol. 1363, Springer-Verlag, Berlin, 1989. With an appendix by J. Baker, O. Slotterbeck and R. Aron. MR 981801, DOI 10.1007/BFb0085267
Jesús M. F. Castillo, Banach spaces, à la recherche du temps perdu, Extracta Math. 15 (2000), no. 2, 373–390. III Congress on Banach Spaces (Jarandilla de la Vera, 1998). MR 1823897
Jesús M. F. Castillo and Manuel González, Three-space problems in Banach space theory, Lecture Notes in Mathematics, vol. 1667, Springer-Verlag, Berlin, 1997. MR 1482801, DOI 10.1007/BFb0112511
Jesús M. F. Castillo, Manuel González, Anatolij M. Plichko, and David Yost, Twisted properties of Banach spaces, Math. Scand. 89 (2001), no. 2, 217–244. MR 1868175, DOI 10.7146/math.scand.a-14339
H. H. Corson, The weak topology of a Banach space, Trans. Amer. Math. Soc. 101 (1961), 1–15. MR 132375, DOI 10.1090/S0002-9947-1961-0132375-5
Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
Joe Diestel, Hans Jarchow, and Andrew Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995. MR 1342297, DOI 10.1017/CBO9780511526138
Per Enflo, Joram Lindenstrauss, and Gilles Pisier, On the “three space problem”, Math. Scand. 36 (1975), no. 2, 199–210. MR 383047, DOI 10.7146/math.scand.a-11571
Ciprian Foiaş and Ivan Singer, On bases in $C([O,\,1])$ and $L^{1}\,([0,\,1])$, Rev. Roumaine Math. Pures Appl. 10 (1965), 931–960. MR 206691
G. Godefroy, N. J. Kalton and G. Lancien, Szlenk indices and uniform homeomorphisms, Trans. Amer. Math. Soc. 353 (2001) 3895–3918.
W. B. Johnson and H. P. Rosenthal, On $\omega ^{\ast }$-basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), 77–92. MR 310598, DOI 10.4064/sm-43-1-77-92
W. B. Johnson and M. Zippin, Separable $L_{1}$ preduals are quotients of $C(\Delta )$, Israel J. Math. 16 (1973), 198–202. MR 338745, DOI 10.1007/BF02757870
W. B. Johnson and M. Zippin, Extension of operators from weak$^*$-closed subspaces of $l_1$ into $C(K)$ spaces, Studia Math. 117 (1995), no. 1, 43–55. MR 1367692
M. I. Kadets and B. Ya. Levin, On a solution of a problem of Banach concerning topological equivalence of spaces of continuous functions (Russian), Trudy Sem. Funct. Anal. Voronezh. 3–4 (1960) 20–25.
N. J. Kalton, On subspaces of $c_0$ and extension of operators into $C(K)$-spaces, Quart. J. Math (Oxford) 52 (2001), 313–328.
N. J. Kalton and N. T. Peck, Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979), 1–30. MR 542869, DOI 10.1090/S0002-9947-1979-0542869-X
N. J. Kalton and A. Pełczyński, Kernels of surjections from $\scr L_1$-spaces with an application to Sidon sets, Math. Ann. 309 (1997), no. 1, 135–158. MR 1467651, DOI 10.1007/s002080050107
H. Knaust, E. Odell, and Th. Schlumprecht, On asymptotic structure, the Szlenk index and UKK properties in Banach spaces, Positivity 3 (1999), no. 2, 173–199. MR 1702641, DOI 10.1023/A:1009786603119
S. Kwapień, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients, Studia Math. 44 (1972), 583–595. MR 341039, DOI 10.4064/sm-44-6-583-595
S. Kwapień, Isomorphic characterizations of Hilbert spaces by orthogonal series with vector valued coefficients, Séminaire Maurey-Schwartz (année 1972–1973), Espaces $L^{p}$ et applications radonifiantes, Centre de Math., École Polytech., Paris, 1973, pp. Exp. No. 8, 7. MR 0410345
J. Lindenstrauss and H. P. Rosenthal, Automorphisms in $c_{0}$, $l_{1}$ and $m$, Israel J. Math. 7 (1969), 227–239. MR 250032, DOI 10.1007/BF02787616
J. Lindenstrauss and H. P. Rosenthal, The ${\cal L}_{p}$ spaces, Israel J. Math. 7 (1969), 325–349. MR 270119, DOI 10.1007/BF02788865
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
Bernard Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces $L^{p}$, Astérisque, No. 11, Société Mathématique de France, Paris, 1974 (French). With an English summary. MR 0344931
Bernard Maurey, Un théorème de prolongement, C. R. Acad. Sci. Paris Sér. A 279 (1974), 329–332 (French). MR 355539
A. Pełczyński, On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular operators in $C(S)$-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 31–36. MR 177300
A. Pełczyński, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Dissertationes Math. (Rozprawy Mat.) 58 (1968), 92. MR 227751
Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99
Haskell P. Rosenthal, Some applications of $p$-summing operators to Banach space theory, Studia Math. 58 (1976), no. 1, 21–43. MR 430749, DOI 10.4064/sm-58-1-21-43
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
B. S. Cirel′son, It is impossible to imbed $\ell _{p}$ or $c_{0}$ into an arbitrary Banach space, Funkcional. Anal. i Priložen. 8 (1974), no. 2, 57–60 (Russian). MR 0350378