Infinite Asymptotic Games and (∗)-Embeddings of Banach Spaces

• Autorzy: Karadakis G. N. I.

Identyfikatory

DOI

10.4064/ba60-2-4

• Języki publikacji: EN

Abstrakty

EN

We use methods of infinite asymptotic games to characterize subspaces of Banach spaces with a finite-dimensional decomposition (FDD) and prove new theorems on operators. We consider a separable Banach space X, a set S of sequences of finite subsets of X and the S-game. We prove that if S satisfies some specific stability conditions, then Player I has a winning strategy in the S-game if and only if X has a skipped-blocking decomposition each of whose skipped-blockings belongs to S. This result implies that if T is a (∗)-embedding of X (a 1-1 operator which maps the balls of subspaces with an FDD to weakly Gδ sets), then, for every n≥4, there exist n subspaces of X with an FDD that generate X and the restriction of T to each of them is a semi-embedding under an equivalent norm. We also prove that X does not contain isomorphic copies of dual spaces if and only if every (∗)-embedding defined on X is an isomorphic embedding. We also deal with the case where X is non-separable, reaching similar results.

Słowa kluczowe

EN

functionally determined; strongly functionally determined sets; skipped-blocking decomposition; semi-embedding; embedding

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2012
• Tom: Vol. 60, no 2
• Strony: 133--154
• Opis fizyczny: Bibliogr. 30 poz.

Twórcy

autor

Karadakis G. N. I.

• 14, Bizaniou Str. 164 51 Argiroupoli Athens, Greece

Bibliografia

• [AGR] S. A. Argyros, G. Godefroy and H. P. Rosenthal, Descriptive set theory and Banach spaces, in: Handbook of the Geometry of Banach Spaces, Vol. 2, W. B. Johnson and J. Lindenstrauss (eds.), North-Holland, Amsterdam, 2003, 1007–1069.
• [AT] S. A. Argyros and S. Todorcevic, Ramsey Methods in Analysis, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser, Basel, 2005.
• [BR1] J. Bourgain and H. P. Rosenthal, Geometrical implications of certain finitedimensional decompositions, Bull. Soc. Math. Belg. 32 (1980), 57–82.
• [BR2] —, —, Applications of the theory of semi-embeddings to Banach space theory, J. Funct. Anal. 52 (1983), 149–188.
• [D] L. Drewnowski, Semi-embeddings of Banach spaces which are hereditarily c0, Proc. Edinburgh Math. Soc. 26 (1983), 163–167.
• [E] P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317.
• [F] V. P. Fonf, Semiimbeddings and Gδ-imbeddings of Banach spaces, Math. Notes 39 (1986), 302–307; transl. from Mat. Zametki 39 (1986), 550–561.
• [FOSZ] D. Freeman, E. Odell, Th. Schlumprecht and A. Zsák, Banach spaces of bounded Szlenk index, II, Fund. Math. 205 (2009), 161–177.
• [GM1] N. Ghoussoub and B. Maurey, Gδ-embeddings in Hilbert space, J. Funct. Anal. 61 (1985), 72–97.
• [GM2] N. Ghoussoub and B. Maurey, Gδ-embeddings in Hilbert space, II, ibid. 78 (1988), 271–305.
• [Go1] W. T. Gowers, A Banach space not containing c0, `1 or a reflexive subspace, Trans. Amer. Math. Soc. 344 (1994), 407–420.
• [Go2] —, An infinite Ramsey theorem and some Banach-space dichotomies, Ann. Of Math. (2) 156 (2002), 797–833.
• [GoM] W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851–874.
• [HHZ] P. Habala, P. Hájek and V. Zizler, Introduction to Banach Spaces, [I], [II], Matfyzpress, Praha, 1996.
• [J] R. C. James, Bases and reflexivity of Banach spaces, Ann. of Math. 52 (1950), 518–527.
• [JZ] W. B. Johnson and B. Zheng, A characterization of subspaces and quotients of reflexive Banach spaces with unconditional bases, Duke Math. J. 141 (2008), 505–518.
• [K1] G.-N. I. Karadakis, Winning strategies for functionally determined properties and their applications to (*)-embeddings of Banach spaces, doctoral thesis, Dept. Math., Univ. of Athens, 2003 (in Greek).
• [K2] G.-N. I. Karadakis, Functionally determined and functionally rejected properties of Banach spaces, preprint.
• [Ke] A. S. Kechris, Classical Descriptive Set Theory, Grad. Texts in Math. 156, Springer, New York, 1995.
• [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Ergeb. Math. Grenzgeb. 92, Springer, Berlin, 1977.
• [LPP] H. P. Lotz, N. T. Peck and H. Porta, Semi-embeddings of Banach spaces, Proc. Edinburgh Math. Soc. 22 (1979), 233–240.
• [MMT] B. Maurey, V. D. Milman and N. Tomczak-Jaegermann, Asymptotic infinitedimensional theory of Banach spaces, in: Oper. Theory Adv. Appl. 77, Birkhäuser, 1994, 149–175.
• [O] E. Odell, On subspaces, asymptotic structure, and distortion of Banach spaces; connections with logic, in: Analysis and Logic, London Math. Soc. Lecture Note Ser. 262, Cambridge Univ. Press, 2002, 189–267.
• [OS1] E. Odell and Th. Schlumprecht, Trees and branches in Banach spaces, Trans. Amer. Math. Soc. 354 (2002), 4085–4108.
• [OS2] —, —, A universal reflexive space for the class of uniformly convex Banach spaces, Math. Ann. 335 (2006), 901–916.
• [OS3] —, —, Embedding into Banach spaces with finite-dimensional decompositions, Rev. R. Acad. Cien. Ser. A Mat. 100 (2006), 295–323.
• [OSZ] E. Odell, Th. Schlumprecht and A. Zsák, A new infinite game in Banach spaces with applications, in: Banach Spaces and their Applications in Analysis, de Gruyter, Berlin, 2007, 147–182.
• [R] C. Rosendal, Infinite asymptotic games, Ann. Inst. Fourier (Grenoble) 59 (2009), 1323–1348.
• [Ro] H. Rosenthal, A characterization of Banach spaces containing `1, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413.
• [Z] B. Zheng, On operators which factor through `p or c0, Studia Math. 176 (2006), 177–190.