The Phillips properties

Authors:
Walden Freedman and Ali Ülger

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2137-2145

MSC (1991):
Primary 46B20, 46B03; Secondary 46L05, 46J10

Published electronically:
February 21, 2000

MathSciNet review:
1766719

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A Banach space has the Phillips property if the canonical projection is sequentially weak-norm continuous, and has the weak Phillips property if is sequentially weak-weak continuous. We study both properties in connection with other geometric properties, such as the Dunford-Pettis property, Pelczynski's properties and (V), and the Schur property.

**[B-De]**J. Bourgain and F. Delbaen,*A class of special \cal𝐿_{∞} spaces*, Acta Math.**145**(1980), no. 3-4, 155–176. MR**590288**, 10.1007/BF02414188**[B-D]**Jean Bourgain and Joe Diestel,*Limited operators and strict cosingularity*, Math. Nachr.**119**(1984), 55–58. MR**774176**, 10.1002/mana.19841190105**[B]**Scott W. Brown,*Weak sequential convergence in the dual of an algebra of compact operators*, J. Operator Theory**33**(1995), no. 1, 33–42. MR**1342475****[C]**Pilar Cembranos,*The hereditary Dunford-Pettis property on 𝐶(𝐾,𝐸)*, Illinois J. Math.**31**(1987), no. 3, 365–373. MR**892175****[Ch]**Cho-Ho Chu,*A note on scattered 𝐶*-algebras and the Radon-Nikodým property*, J. London Math. Soc. (2)**24**(1981), no. 3, 533–536. MR**635884**, 10.1112/jlms/s2-24.3.533**[D1]**Joseph Diestel,*Sequences and series in Banach spaces*, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR**737004****[D2]**Joe Diestel,*A survey of results related to the Dunford-Pettis property*, Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979) Contemp. Math., vol. 2, Amer. Math. Soc., Providence, R.I., 1980, pp. 15–60. MR**621850****[Di]**Jacques Dixmier,*𝐶*-algebras*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett; North-Holland Mathematical Library, Vol. 15. MR**0458185****[GGMS]**N. Ghoussoub, G. Godefroy, B. Maurey, and W. Schachermayer,*Some topological and geometrical structures in Banach spaces*, Mem. Amer. Math. Soc.**70**(1987), no. 378, iv+116. MR**912637**, 10.1090/memo/0378**[HWW]**P. Harmand, D. Werner, and W. Werner,*𝑀-ideals in Banach spaces and Banach algebras*, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993. MR**1238713****[HOR]**R. Haydon, E. Odell, and H. Rosenthal,*On certain classes of Baire-1 functions with applications to Banach space theory*, Functional analysis (Austin, TX, 1987/1989) Lecture Notes in Math., vol. 1470, Springer, Berlin, 1991, pp. 1–35. MR**1126734**, 10.1007/BFb0090209**[K-O]**H. Knaust and E. Odell,*On 𝑐₀ sequences in Banach spaces*, Israel J. Math.**67**(1989), no. 2, 153–169. MR**1026560**, 10.1007/BF02937292**[L-U]**Anthony To Ming Lau and Ali Ülger,*Some geometric properties on the Fourier and Fourier-Stieltjes algebras of locally compact groups, Arens regularity and related problems*, Trans. Amer. Math. Soc.**337**(1993), no. 1, 321–359. MR**1147402**, 10.1090/S0002-9947-1993-1147402-7**[P]**A. Pełczyński,*Banach spaces on which every unconditionally converging operator is weakly compact*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**10**(1962), 641–648. MR**0149295****[P-S]**A. Pełczyński and Z. Semadeni,*Spaces of continuous functions. III. Spaces 𝐶(Ω) for Ω without perfect subsets*, Studia Math.**18**(1959), 211–222. MR**0107806****[P-Sz]**A. Pełczyński and W. Szlenk,*An example of a non-shrinking basis*, Rev. Roumaine Math. Pures Appl.**10**(1965), 961–966. MR**0203432****[Pf]**H. Pfitzner,*Weak compactness in the dual of a 𝐶*-algebra is determined commutatively*, Math. Ann.**298**(1994), no. 2, 349–371. MR**1256621**, 10.1007/BF01459739**[R]**Haskell Rosenthal,*A characterization of Banach spaces containing 𝑐₀*, J. Amer. Math. Soc.**7**(1994), no. 3, 707–748. MR**1242455**, 10.1090/S0894-0347-1994-1242455-4**[U]**Ali Ülger,*Subspaces and subalgebras of 𝐾(𝐻) whose duals have the Schur property*, J. Operator Theory**37**(1997), no. 2, 371–378. MR**1452283**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
46B20,
46B03,
46L05,
46J10

Retrieve articles in all journals with MSC (1991): 46B20, 46B03, 46L05, 46J10

Additional Information

**Walden Freedman**

Affiliation:
Department of Mathematics, College of Arts and Sciences, Koç University, 80860 Istinye, Istanbul, Turkey

Email:
wfreedman@ku.edu.tr

**Ali Ülger**

Affiliation:
Department of Mathematics, College of Arts and Sciences, Koç University, 80860 Istinye, Istanbul, Turkey

Email:
aulger@ku.edu.tr

DOI:
https://doi.org/10.1090/S0002-9939-00-05703-8

Keywords:
Phillips lemma,
Dunford-Pettis property,
property $(u)$,
property (V),
Schur property

Received by editor(s):
September 7, 1998

Published electronically:
February 21, 2000

Communicated by:
Dale Alspach

Article copyright:
© Copyright 2000
by Walden Freedman and Ali Ülger