C*-algebras have a quantitative version of Pełczyński's property (V)

Hana Krulišová

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 4, page 937-951
  • ISSN: 0011-4642

Abstract

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A Banach space X has Pełczyński’s property (V) if for every Banach space Y every unconditionally converging operator T : X → Y is weakly compact. H. Pfitzner proved that C * -algebras have Pełczyński’s property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that C ( K ) spaces for a compact Hausdorff space K enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover, we prove that in dual Banach spaces a quantitative version of the property (V) implies a quantitative version of the Grothendieck property.

How to cite

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Krulišová, Hana. "C*-algebras have a quantitative version of Pełczyński's property (V)." Czechoslovak Mathematical Journal 67.4 (2017): 937-951. <http://eudml.org/doc/294226>.

@article{Krulišová2017,
abstract = {A Banach space $X$ has Pełczyński’s property (V) if for every Banach space $Y$ every unconditionally converging operator $T\colon X\rightarrow Y$ is weakly compact. H. Pfitzner proved that $C^*$-algebras have Pełczyński’s property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that $C(K)$ spaces for a compact Hausdorff space $K$ enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover, we prove that in dual Banach spaces a quantitative version of the property (V) implies a quantitative version of the Grothendieck property.},
author = {Krulišová, Hana},
journal = {Czechoslovak Mathematical Journal},
keywords = {Pełczyński’s property (V); $C^*$-algebra; Grothendieck property},
language = {eng},
number = {4},
pages = {937-951},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {C*-algebras have a quantitative version of Pełczyński's property (V)},
url = {http://eudml.org/doc/294226},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Krulišová, Hana
TI - C*-algebras have a quantitative version of Pełczyński's property (V)
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 4
SP - 937
EP - 951
AB - A Banach space $X$ has Pełczyński’s property (V) if for every Banach space $Y$ every unconditionally converging operator $T\colon X\rightarrow Y$ is weakly compact. H. Pfitzner proved that $C^*$-algebras have Pełczyński’s property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that $C(K)$ spaces for a compact Hausdorff space $K$ enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover, we prove that in dual Banach spaces a quantitative version of the property (V) implies a quantitative version of the Grothendieck property.
LA - eng
KW - Pełczyński’s property (V); $C^*$-algebra; Grothendieck property
UR - http://eudml.org/doc/294226
ER -

References

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  1. Angosto, C., Cascales, B., 10.1016/j.topol.2008.12.011, Topology Appl. 156 (2009), 1412-1421. (2009) Zbl1176.46012MR2502017DOI10.1016/j.topol.2008.12.011
  2. Behrends, E., New proofs of Rosenthal’s ℓ 1 -theorem and the Josefson-Nissenzweig theorem, Bull. Pol. Acad. Sci., Math. 43 (1995), 283-295. (1995) Zbl0847.46007MR1414785
  3. Bendová, H., 10.1016/j.jmaa.2013.11.033, J. Math. Anal. Appl. 412 (2014), 1097-1104. (2014) Zbl1322.46008MR3147271DOI10.1016/j.jmaa.2013.11.033
  4. Blasi, F. S. De, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Sér. 21 (1977), 259-262. (1977) Zbl0365.46015MR0482402
  5. Gasparis, I., 10.1016/j.jmaa.2015.09.079, J. Math. Anal. Appl. 434 (2016), 1160-1165. (2016) Zbl06509536MR3415714DOI10.1016/j.jmaa.2015.09.079
  6. Harmand, P., Werner, D., Werner, W., 10.1007/BFb0084355, Lecture Notes in Mathematics 1547, Springer, Berlin (1993). (1993) Zbl0789.46011MR1238713DOI10.1007/BFb0084355
  7. Kalenda, O. F. K., Pfitzner, H., Spurný, J., 10.1016/j.jfa.2011.02.006, J. Funct. Anal. 260 (2011), 2986-2996. (2011) Zbl1248.46012MR2774062DOI10.1016/j.jfa.2011.02.006
  8. Krulišová, H., Quantification of Pe{ł}czyński's property (V), To appear in Math. Nachr. 
  9. Lechner, J., 10.1016/j.jmaa.2016.06.038, J. Math. Anal. Appl. 446 (2017), 1362-1371. (2017) Zbl1364.46015MR3563039DOI10.1016/j.jmaa.2016.06.038
  10. Pfitzner, H., 10.1007/BF01459739, Math. Ann. 298 (1994), 349-371. (1994) Zbl0791.46035MR1256621DOI10.1007/BF01459739
  11. Rudin, W., Real and Complex Analysis, McGraw-Hill, New York (1987). (1987) Zbl0925.00005MR0924157
  12. Simons, S., 10.1007/BF01430962, Math. Ann. 216 (1975), 225-231. (1975) Zbl0294.46010MR0402470DOI10.1007/BF01430962
  13. Takesaki, M., Theory of Operator Algebras I, Encyclopaedia of Mathematical Sciences 124, Operator Algebras and Non-Commutative Geometry 5, Springer, Berlin (2002). (2002) Zbl0990.46034MR1873025

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