Accessible operators on ultraproducts of Banach spaces
DOI:
https://doi.org/10.17398/2605-5686.39.2.207Palabras clave:
Ultraproduct, quasi-Banach space, operator, exact sequence, quasilinear map, K-spaceResumen
We address a question by Henry Towsner about the possibility of representing linear operators between ultraproducts of Banach spaces by means of ultraproducts of nonlinear maps.
We provide a bridge between these “accessible” operators and the theory of twisted sums through the so-called quasilinear maps. Thus, for many pairs of Banach spaces X and Y , there is an “accessible” operator XU → YU that is not the ultraproduct of a family of operators X → Y if and only if there is a short exact sequence of quasi-Banach spaces and operators 0 → Y → Z → X → 0 that does not split. We then adapt classical work by Ribe and Kalton–Peck to exhibit pretty concrete examples of accessible functionals and endomorphisms for the sequence spaces lp
The paper is organized so that the main ideas are accessible to readers working on ultraproducts and requires only a rustic knowledge of Banach space theory.
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W.G. Bade, P.C. Curtis Jr., H.G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc.
(3) 55 (1987), 359 – 377.
F. Cabello Sánchez, A simple proof that super-reflexive spaces are K-spaces, Proc. Amer. Math. Soc. 132 (3) (2004), 697 – 698.
F. Cabello Sánchez, Nonlinear centralizers in homology II. The Schatten classes, Rev. Mat. Iberoam. 37 (6) (2021), 2309 – 2346.
F. Cabello Sánchez, J.M.F. Castillo, Stability constants and the homology of quasi-Banach spaces, Israel J. Math. 198 (1) (2013), 347 – 370.
F. Cabello Sánchez, J.M.F. Castillo, “Homological methods in Banach space theory”, Cambridge Stud. Adv. Math. 203, Cambridge University Press, Cambridge, 2023.
F. Cabello Sánchez, J.M.F. Castillo, J. Suárez, On strictly singular nonlinear centralizers, Nonlinear Anal. 75 (7) (2012), 3313 – 3321,
F. Cabello Sánchez, J. Garbulińska-Wȩgrzyn, W. Kubiś, Quasi-Banach spaces of almost universal disposition, J. Funct. Anal. 267 (3) (2014), 744 – 771.
M. Daws, Amenability of ultrapowers of Banach algebras, Proc. Edinb. Math. Soc. (2) 52 (2009), 307 – 338. Corrigendum: Proc. Edinb. Math. Soc. (2) 53 (2010), 633 – 637.
P. Enflo, J. Lindenstrauss, G. Pisier, On the three space problem, Math. Scand. 36 (1975), 199 – 210.
S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72 – 104.
C.W. Henson, J. Iovino, Ultraproducts in analysis, in “Analysis and logic (Mons,1997)”, London Math. Soc. Lecture Note Ser., 262, Cambridge University Press, Cambridge, 2002, 1 – 110.
R.C. James, Bases and reflexivity of Banach spaces, Ann. of Math. (2) 52 (1950), 518 – 527.
R.C. James, A non-reflexive Banach space isometric with its second conjugate space, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 174 – 177.
N.J. Kalton, The three space problem for locally bounded F-spaces, Compositio Math. 37 (1978), 243 – 276.
N.J. Kalton, Transitivity and quotients of Orlicz spaces, Comment. Math. Spec. Issue 1 (1978), 159 – 172.
N.J. Kalton, Locally complemented subspaces and Lp -spaces for 0 < p < 1, Math. Nachr. 115 (1984), 71 – 97.
N.J. Kalton, The Maharam problem, in “Séminaire d’Initiation à l’Analyse”, Exp. No. 18, 13 pp., Publ. Math. Univ. Pierre et Marie Curie, 94, Université de Paris VI, Paris, 1989.
N.J. Kalton, Quasi-Banach spaces, in “Handbook of the geometry of Banach spaces, Vol. 2, (Edited by W. B. Johnson and J. Lindenstrauss), North-Holland, Amsterdam, 2003, 1099 – 1130.
N.J. Kalton, M. Ostrovskii, Distances between Banach spaces, Forum Math. 11 (1999), 17 – 48.
N.J. Kalton, N.T. Peck, Twisted sums of sequence spaces and the three-space problem, Trans. Amer. Math. Soc. 255 (1979), 1 – 30.
N.J. Kalton, N.T. Peck, J.W. Roberts, “An F -space sampler”, London Math. Soc. Lecture Note Ser., 89, Cambridge University Press, Cambridge, 1984.
N.J. Kalton, J.W. Roberts, Uniformly exhaustive submeasures and nearly additive set functions, Trans. Amer. Math. Soc. 278 (1983), 803 – 816.
G. Pisier, Counterexamples to a conjecture of Grothendieck, Acta Math. 151 (3-4) (1983), 181 – 208.
G. Pisier, H. Xu, Non-commutative Lp -spaces, in “Handbook of the geometry of Banach spaces, Vol. 2, (Edited by W. B. Johnson and J. Lindenstrauss), North-Holland, Amsterdam, 2003, 1459 – 1517.
M. Ribe, Examples for the nonlocally convex three space problem, Proc. Amer. Math. Soc. 73 (1979), 351 – 355.
V. Runde, “Amenable Banach algebra. A panorama”, Springer Monogr. Math., Springer-Verlag, New York, 2020.
B. Sims, ““Ultra”-techniques in Banach space theory”, Queen’s Papers in Pure and Appl. Math., 60, Queen’s University, Kingston, ON, 1982. (At least two later versions exist.)
H. Towsner, https://mathoverflow.net/questions/53260/dual-of-the-ultraproduct-of-a-banach-space
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Esta obra está bajo una licencia internacional Creative Commons Atribución-NoComercial 4.0.
