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ON A FUNCTION MODULE WITH APPROXIMATE HYPERPLANE SERIES PROPERTY

Part of: Linear function spaces and their duals Basic linear algebra

Published online by Cambridge University Press:  03 July 2019

T. GRANDO Open the ORCID record for T. GRANDO [Opens in a new window]  and
M. L. LOURENÇO
T. GRANDO*
Affiliation:
Department of Mathematics, University of São Paulo, P.O. Box 66281, 05315-970, São Paulo, Brazil email tgrando@ime.usp.br
M. L. LOURENÇO
Affiliation:
Department of Mathematics, University of São Paulo, P.O. Box 66281, 05315-970, São Paulo, Brazil email mllouren@ime.usp.br
*
✉tgrando@ime.usp.br
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Abstract

We present a sufficient and necessary condition for a function module space $X$ to have the approximate hyperplane series property (AHSP). As a consequence, we have that the space ${\mathcal{C}}_{0}(L,E)$ of bounded and continuous $E$-valued mappings defined on the locally compact Hausdorff space $L$ has AHSP if and only if $E$ has AHSP.

Keywords

Bishop–Phelps–Bollobás property for operatorsapproximate hyperplane series propertyfunction module space

MSC classification

Primary: 46E25: Rings and algebras of continuous, differentiable or analytic functions
Secondary: 15A03: Vector spaces, linear dependence, rank
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 108 , Issue 3 , June 2020 , pp. 341 - 348
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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References

Acosta, M. D., Aron, R. M., García, D. and Maestre, M., ‘The Bishop–Phelps–Bollobás theorem for operators’, J. Funct. Anal. 254 (2008), 2780–2799.Google Scholar
Acosta, M. D., Becerra-Guerrero, J., Choi, Y. S., Ciesielski, M., Kim, S. K., Lee, H. J., Lourenço, M. L. and Martín, M., ‘The Bishop–Phelps–Bollobás property for operators between spaces of continuous functions’, Nonlinear Anal. 95 (2014), 323–332.Google Scholar
Acosta, M. D., Becerra-Guerrero, J., García, D., Kim, S. K. and Maestre, M., ‘Bishop–Phelps–Bollobás property for certain spaces of operators’, J. Math. Anal. Appl. 414 (2014), 532–545.Google Scholar
Aron, R. M., Cascales, B. and Kozhushkina, O., ‘The Bishop–Phelps–Bollobás theorem and Asplund operators’, Proc. Amer. Math. Soc. 139(10) (2011), 3553–3560.Google Scholar
Aron, R. M., Choi, Y. S., García, D. and Maestre, M., ‘The Bishop–Phelps–Bollobás theorem for 𝓛(L 1(𝜇), L ∞[0, 1])’, Adv. Math. 228(1) (2011), 617–628.Google Scholar
Aron, R. M., Choi, Y. S., Kim, S. K., Lee, H. J. and Martín, M., ‘The Bishop–Phelps–Bollobás version of Lindenstrauss properties A and B’, Trans. Amer. Math. Soc. 367 (2015), 6085–6101.Google Scholar
Behrends, E., M-Structure and the Banach–Stone Theorem (Springer, Berlin, 1979).Google Scholar
Bishop, E. and Phelps, R. R., ‘A proof that every Banach space is subreflexive’, Bull. Amer. Math. Soc. (N.S.) 67 (1961), 97–98.Google Scholar
Bollobás, B., ‘An extension to the theorem of Bishop and Phelps’, Bull. Lond. Math. Soc. 2 (1970), 181–182.Google Scholar
Cascales, B., Guirao, A. J. and Kadets, V., ‘A Bishop–Phelps–Bollobás type theorem for uniform algebras’, Adv. Math. 240 (2013), 370–382.Google Scholar
Chica, M., Kadets, V., Martín, M., Moreno-Pulido, S. and Rambla-Barreno, F., ‘Bishop–Phelps–Bollobás moduli of a Banach space’, J. Math. Anal. Appl. 412(2) (2014), 697–719.Google Scholar
Choi, Y. S. and Kim, S. K., ‘The Bishop–Phelps–Bollobás theorem for operators from L 1(𝜇) to Banach spaces with the Radon–Nikodým property’, J. Funct. Anal. 261(6) (2011), 1446–1456.Google Scholar
Choi, Y. S. and Kim, S. K., ‘The Bishop–Phelps–Bollobás property and lush spaces’, J. Math. Anal. Appl. 390 (2012), 549–555.Google Scholar
Choi, Y. S., Kim, S. K., Lee, H. J. and Martín, M., ‘The Bishop–Phelps–Bollobás theorem for operators on L 1(𝜇)’, J. Funct. Anal. 267(91) (2014), 214–242.Google Scholar
Choi, Y. S., Kim, S. K., Lee, H. J. and Martín, M., ‘On Banach spaces with the approximate hyperplane series property’, Banach J. Math. Anal. 9(4) (2015), 243–258.Google Scholar
Kim, S. K., ‘The Bishop–Phelps–Bollobás theorem for operators from c 0 to uniformly convex spaces’, Israel J. Math. 197(1) (2013), 425–435.Google Scholar
Kim, S. K. and Lee, H. J., ‘Uniform convexity and Bishop–Phelps–Bollobás property’, Canad. J. Math. 66(2) (2014), 372–386.Google Scholar
Lindenstrauss, J., ‘On operators which attain their norm’, Israel J. Math. 1 (1963), 139–148.Google Scholar