On decompositions of Banach spaces of continuous functions on Mrówka's spaces
Author:
Piotr Koszmider
Journal:
Proc. Amer. Math. Soc. 133 (2005), 21372146
MSC (2000):
Primary 03E50, 46E15, 54G12
Published electronically:
February 25, 2005
MathSciNet review:
2137881
Fulltext PDF Free Access
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Abstract: It is well known that if is infinite compact Hausdorff and scattered (i.e., with no perfect subsets), then the Banach space of continuous functions on has complemented copies of , i.e., . We address the question if this could be the only type of decompositions of into infinitedimensional summands for infinite, scattered. Making a special settheoretic assumption such as the continuum hypothesis or Martin's axiom we construct an example of Mrówka's space (i.e., obtained from an almost disjoint family of sets of positive integers) which answers positively the above question.
 1.
D.
Alspach and Y.
Benyamini, Primariness of spaces of continuous functions on
ordinals, Israel J. Math. 27 (1977), no. 1,
64–92. MR
0440349 (55 #13224)
 2.
Joseph
Diestel, Sequences and series in Banach spaces, Graduate Texts
in Mathematics, vol. 92, SpringerVerlag, New York, 1984. MR 737004
(85i:46020)
 3.
Eric
K. van Douwen, The integers and topology, Handbook of
settheoretic topology, NorthHolland, Amsterdam, 1984,
pp. 111–167. MR 776622
(87f:54008)
 4.
W.
T. Gowers and B.
Maurey, The unconditional basic sequence
problem, J. Amer. Math. Soc.
6 (1993), no. 4,
851–874. MR 1201238
(94k:46021), http://dx.doi.org/10.1090/S08940347199312012380
 5.
W.
B. Johnson and J.
Lindenstrauss, Some remarks on weakly compactly generated Banach
spaces, Israel J. Math. 17 (1974), 219–230. MR 0417760
(54 #5808)
W.
B. Johnson and J.
Lindenstrauss, Correction to: “Some remarks on weakly
compactly generated Banach spaces” [Israel J. Math. 17 (1974),
219–230;\ MR 54 #5808], Israel J. Math. 32
(1979), no. 4, 382–383. MR 571092
(81g:46015), http://dx.doi.org/10.1007/BF02760467
 6.
P. Koszmider; Banach spaces of large densities but few operators. Preprint.
 7.
Piotr
Koszmider, Banach spaces of continuous functions with few
operators, Math. Ann. 330 (2004), no. 1,
151–183. MR 2091683
(2005h:46027), http://dx.doi.org/10.1007/s002080040544z
 8.
Kenneth
Kunen, Set theory, Studies in Logic and the Foundations of
Mathematics, vol. 102, NorthHolland Publishing Co., AmsterdamNew
York, 1980. An introduction to independence proofs. MR 597342
(82f:03001)
 9.
J.
Lindenstrauss and A.
Pełczyński, Contributions to the theory of the
classical Banach spaces, J. Functional Analysis 8
(1971), 225–249. MR 0291772
(45 #863)
 10.
N.
N. Luzin, On subsets of the series of natural numbers,
Izvestiya Akad. Nauk SSSR. Ser. Mat. 11 (1947),
403–410 (Russian). MR 0021576
(9,82c)
 11.
A. Miller; private notes, 2003.
 12.
Aníbal
Moltó, On a theorem of Sobczyk, Bull. Austral. Math.
Soc. 43 (1991), no. 1, 123–130. MR 1086724
(92d:46043), http://dx.doi.org/10.1017/S0004972700028835
 13.
S.
Mrówka, Some settheoretic constructions in topology,
Fund. Math. 94 (1977), no. 2, 83–92. MR 0433388
(55 #6364)
 14.
A.
Pełczyński, Projections in certain Banach
spaces, Studia Math. 19 (1960), 209–228. MR 0126145
(23 #A3441)
 15.
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
(21 #6528)
 16.
Haskell
P. Rosenthal, On relatively disjoint families of measures, with
some applications to Banach space theory, Studia Math.
37 (1970), 13–36. MR 0270122
(42 #5015)
 17.
Zbigniew
Semadeni, Banach spaces of continuous functions. Vol. I,
PWN—Polish Scientific Publishers, Warsaw, 1971. Monografie
Matematyczne, Tom 55. MR 0296671
(45 #5730)
 18.
Saharon
Shelah, A Banach space with few operators, Israel J. Math.
30 (1978), no. 12, 181–191. MR 508262
(80b:46033), http://dx.doi.org/10.1007/BF02760838
 19.
Saharon
Shelah and Juris
Steprāns, A Banach space on which there are few
operators, Proc. Amer. Math. Soc.
104 (1988), no. 1,
101–105. MR
958051 (90a:46047), http://dx.doi.org/10.1090/S00029939198809580519
 20.
H.
M. Wark, A nonseparable reflexive Banach space on which there are
few operators, J. London Math. Soc. (2) 64 (2001),
no. 3, 675–689. MR 1865556
(2003a:46031), http://dx.doi.org/10.1112/S0024610701002393
 1.
 D. Alspach, Y. Benyamini; Primariness of spaces of continuous functions on ordinals; Israel J. Math. vol 27, No 1, 1977, pp. 64  92. MR 0440349 (55:13224)
 2.
 J. Diestel; Sequences and series in Banach spaces; SpringerVerlag 1984. MR 0737004 (85i:46020)
 3.
 E. van Douwen; The Integers and Topology; Handbook of Settheoretic Topology; eds K. Kunen, J. Vaughan, North Holland 1984; pp. 111  167. MR 0776622 (87f:54008)
 4.
 W. T. Gowers, B. Maurey; The unconditional basic sequence problem; Journal A. M. S. 6 (1993), pp. 851874. MR 1201238 (94k:46021)
 5.
 W. Johnson, J. Lindenstrauss; Some remarks on weakly compactly generated Banach spaces; Israel J. Math. 17, 1974, pp. 219  230. and Israel J. Math. 32 (1979), no. 4, pp. 382  383. MR 0417760 (54:5808); MR 0571092 (81g:46015)
 6.
 P. Koszmider; Banach spaces of large densities but few operators. Preprint.
 7.
 P. Koszmider; Banach spaces of continuous functions with few operators; Math. Annalen. 330, 2004, pp. 151  183. MR 2091683
 8.
 K. Kunen; Set Theory. An Introduction to Independence Proofs; North Holland, 1980. MR 0597342 (82f:03001)
 9.
 Y. Lindenstrauss, A. Peczynski; Contributions to the theory of the classical Banach spaces; J. Funct. Anal. 8, 1971, pp. 225  249. MR 0291772 (45:863)
 10.
 N. Luzin; On subsets of the series of natural numbers; Izv. Akad. Nauk SSSR, Ser. Mat.., 11, pp. 403  411. MR 0021576 (9:82c)
 11.
 A. Miller; private notes, 2003.
 12.
 A. Moltó; On a theorem of Sobczyk; Bull. Austral. Math. Soc. 43, 1991, 123  130. MR 1086724 (92d:46043)
 13.
 S. Mrówka; Some settheoretic constructions in topology; Fund. Math. vol. XCIV, 1977, pp. 83  92. MR 0433388 (55:6364)
 14.
 A. Peczynski; Projections in certain Banach spaces; Studia Math. vol. XIX, 1960, pp. 209  228. MR 0126145 (23:A3441)
 15.
 A. Peczynski, Z. Semadeni; Spaces of continuous functions (III) (Spaces for without perfect subsets); Studia Math. 18, 1959, pp. 211  222. MR 0107806 (21:6528)
 16.
 H. Rosenthal; On relatively disjoint families of measures with some applications to Banach space theory; Studia Math. 37, (1970), pp. 1336. MR 0270122 (42:5015)
 17.
 Z. Semadeni; Banach spaces of continuous functions; Panstwowe Wydawnictwo Naukowe, 1971. MR 0296671 (45:5730)
 18.
 S. Shelah; A Banach space with few operators; Israel J. Math. 30 (1978), pp. 181191. MR 0508262 (80b:46033)
 19.
 S. Shelah, J. Steprans; A Banach space on which there are few operators; Proc. Amer. Math. Soc. 104 (1988), pp. 101105. MR 0958051 (90a:46047)
 20.
 H. Wark; A nonseparable reflexive Banach space on which there are few operators. J. London Math. Soc. (2) 64 (2001), no. 3, pp. 675  689. MR 1865556 (2003a:46031)
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Additional Information
Piotr Koszmider
Affiliation:
Departamento de Matemática, Universidade de São Paulo, Caixa Postal: 66281, São Paulo, Sp CEP: 05315970, Brazil
Email:
piotr@ime.usp.br
DOI:
http://dx.doi.org/10.1090/S0002993905077993
PII:
S 00029939(05)077993
Keywords:
Banach spaces of continuous functions,
few operators,
scattered spaces,
almost disjoint families
Received by editor(s):
July 24, 2003
Received by editor(s) in revised form:
April 15, 2004
Published electronically:
February 25, 2005
Additional Notes:
The author acknowledges support from CNPQ, Processo Número 300369/018, from FAPESP, Processo Número 02/036777 and from Centre de Recerca Matemática at Universidad Autonoma de Barcelona.
Communicated by:
Alan Dow
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
