Atlas: Three-space problems for polynomial properties in Banach spaces by Francisco Arranz

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Functional Analysis Valencia 2000
July 3-7, 2000
Technical University of Valencia (UPV) and University of Valencia (UV)
Valencia, Spain

Organizers
R.M. Aron (Kent State U., USA), K.D. Bierstedt (U. Paderborn, Germany), J. Bonet (UPV), J. Cerdà (U. Barcelona, Spain), H. Jarchow (U. Zürich, Switzerland), M. Maestre (UV), J. Schmets (U. Liège, Belgium)

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Three-space problems for polynomial properties in Banach spaces
by
Francisco Arranz
Departamento de Matemáticas, Universidad de Extremadura
Coauthors: Jesus M.F. Castillo, Ricardo Garcia

The purpuse of this talk is to answer some questions posed in the monograph [3, Appendix 4.20]. Precisely, the authors complain there the lack of "three-space" results for polynomial properties. Let us recall that if P and Q are two properties of Banach spaces stable by isomorphisms then a Banach space X is said to have the P-by-Q property if it admits a subspace Y with property P so that X/Y has property Q.

A property P is said to be a three-space property if P-by-Q implies P. If a Banach space has simultaneously properties P and Q we shall say that it has the property P&Q property. It will be shown in this talk that the following are not 3-space properties:

The polynomial (RP) of Aron, Choi and Llavona.

The polynomial Schur property of Carne, Cole and Gamelin.

The polynomial Dunford-Pettis property of Gonzalez and Gutierrez.

All this together with other results exposed in the talk, provides a fairly complete catalog of the current state of 3-space problems related to polynomial properties. In the last part of the talk we observe that all couterexamples to the 3-space for properties M_n have in common the presence of copies of l₁. On the other hand, no available method can produce a counterexample made with reflexive spaces. In between, we shall consider "not containing l₁" as a weak form of reflexivity and will tackle the 3-space problem for the property M₂ not containing l₁; or, equivalently, L(X, X^*) = K(X, X^*). New examples of such M₂, such as the Johnson-Lindenstrauss spaces, will be shown.

REFERENCES

[1] R. Aron, Y. Choi and J. Llavona, Estimates by polynomials, Bull. Austral. Math. Soc. 52 (3)(1995), 475-486.

[2] T. Carne, B. Cole and T. Gamelin, A uniform algebra of analytic functions an on Banach space, Trans. Amer. Math. Soc. 314 (1989), 639-659.

[3] J. M. F. Castillo, M. González, Three-space problems in Banach spaces theory, Lecture Notes in Math. 1667, Springer-Verlag, Berlin-Heidelberg-New York, 1997.

[4] W. B. Johnson and J. Lindenstrauss, Some remarks on weakly compactly generated Banach space, Israel J. Math. 17 (1974), 219-230.

[5] N. J. Kalton and N. T. Peck, Twisted sums of sequence spaces and three space problem, Trans. Amer. Math. Soc. 255 (1979), 1-30.

(T)

Date received: April 17, 2000