×

Sailing over three problems of Koszmider. (English) Zbl 1455.46028

Summary: We discuss three problems of P. Koszmider [Proc. Am. Math. Soc. 133, No. 7, 2137–2146 (2005; Zbl 1085.46015)] on the structure of the spaces of continuous functions on the Stone compact \(K_{\mathcal{A}}\) generated by an almost disjoint family \(\mathcal{A}\) of infinite subsets of \(\omega \) – we present a solution to two problems and develop previous results of W. Marciszewski and R. Pol answering the third one [J. Math. Anal. Appl. 350, No. 2, 708–722 (2009; Zbl 1167.46311)]. We will show, in particular, that assuming Martin’s axiom, the space \(C( K_{\mathcal{A}})\) is uniquely determined up to isomorphism by the cardinality of \(\mathcal{A}\) whenever \(| \mathcal{A} | < \mathfrak{c} \), while there are \(2^{\mathfrak{c}}\) nonisomorphic spaces \(C( K_{\mathcal{A}})\) with \(| \mathcal{A} | = \mathfrak{c} \). We also investigate Koszmider’s problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of \(c_0\) and some \(C(K)\) spaces.

MSC:

46E15 Banach spaces of continuous, differentiable or analytic functions
03E50 Continuum hypothesis and Martin’s axiom
54G12 Scattered spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Argyros, S.; Raikoftsalis, T., Banach spaces with a unique nontrivial decomposition, Proc. Am. Math. Soc., 136, 3611-3620 (2008) · Zbl 1162.46011
[2] Avilés, A.; Cabello Sánchez, F.; Castillo, J. M.F.; González, M.; Moreno, Y., Separably Injective Banach Spaces, Lecture Notes in Mathematics, vol. 2132 (2016), Springer · Zbl 1379.46002
[3] Avilés, A.; Marciszewski, W.; Plebanek, G., Twisted sums of \(c_0\) and \(C(K)\)-spaces: a solution to the CCKY problem, Advances in Math. (to appear) · Zbl 1445.46015
[4] Bourgain, J.; Fremlin, D. H.; Talagrand, M., Pointwise compact sets of Baire-measurable functions, Am. J. Math., 100, 845-886 (1978) · Zbl 0413.54016
[5] Cabello Sánchez, F.; Castillo, J. M.F., The long homology sequence for quasi Banach spaces, with applications, Positivity, 8, 379-394 (2004) · Zbl 1086.46046
[6] Cabello Sánchez, F.; Castillo, J. M.F., Homological Methods in Banach Space Theory, Cambridge Studies in Advanced Math. (2020), Cambridge Univ. Press. Scheduled
[7] Cabello Sánchez, F.; Castillo, J. M.F.; Yost, D., Sobczyk’s theorems from A to B, Extr. Math., 15, 391-420 (2000) · Zbl 0990.46003
[8] Cabello Sánchez, F.; Castillo, J. M.F.; Kalton, N. J.; Yost, D. T., Twisted sums with \(C(K)\) spaces, Trans. Am. Math. Soc., 355, 4523-4541 (2003) · Zbl 1066.46006
[9] Castillo, J. M.F.; Papini, P. L., Epheastus account on Trojanski’s polyhedral war, Extr. Math., 29, 35-51 (2014) · Zbl 1328.46008
[10] Castillo, J. M.F., Nonseparable \(C(K)\)-spaces can be twisted when K is a finite height compact, Topol. Appl., 198, 107-116 (2016) · Zbl 1366.46060
[11] Castillo, J. M.F.; González, M., Three-Space Problems in Banach Space Theory, Lecture Notes in Math., vol. 1667 (1997), Springer · Zbl 0914.46015
[12] Castillo, J. M.F.; Moreno, Y., On the Lindenstrauss-Rosenthal theorem, Isr. J. Math., 140, 253-270 (2004) · Zbl 1063.46003
[13] Castillo, J. M.F.; Simoes, M., Property (V) still fails the 3-space property, Extr. Math., 5-11 (2012) · Zbl 1282.46013
[14] Correa, C., Nontrivial twisted sums for finite height spaces under Martin’s Axiom, Fundam. Math., 248, 195-204 (2020) · Zbl 1445.46027
[15] Correa, C.; Tausk, D. V., Nontrivial twisted sums of \(c_0\) and \(C(K)\), J. Funct. Anal., 270, 842-853 (2016) · Zbl 1347.46014
[16] Correa, C.; Tausk, D. V., Local extension property for finite height spaces, Fundam. Math., 245, 149-165 (2019) · Zbl 1472.06017
[17] Debs, G., Descriptive aspects of Rosenthal compacta, (Recent Progress in General Topology III (2014), Atlantis Press: Atlantis Press Paris), 205-227 · Zbl 1305.54001
[18] Dobrowolski, T.; Marciszewski, W., Classification of function spaces with the pointwise topology determined by a countable dense set, Fundam. Math., 148, 35-62 (1995) · Zbl 0834.46016
[19] Dow, A.; Vaughan, J. E., Mrówka maximal almost disjoint families for uncountable cardinals, Topol. Appl., 157, 1379-1394 (2010) · Zbl 1196.54060
[20] Godefroy, G., Compacts de Rosenthal, Pac. J. Math., 91, 293-306 (1980) · Zbl 0475.46003
[21] Hrušák, M., Almost disjoint families and topology, (Recent Progress in General Topology III (2014), Atlantis Press: Atlantis Press Paris), 601-638 · Zbl 1326.54003
[22] Hernández-Hernández, F.; Hrušák, M., Topology of Mrówka-Isbell spaces, (Pseudocompact Topological Spaces. Pseudocompact Topological Spaces, Dev. Math., vol. 55 (2018), Springer: Springer Cham), 253-289 · Zbl 1447.54006
[23] Johnson, W. B.; Lindenstrauss, J., Some remarks on weakly compactly generated Banach spaces, Isr. J. Math., 17, 219-230 (1974) · Zbl 0306.46021
[24] Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156 (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0819.04002
[25] Koszmider, P., On decomposition of Banach spaces of continuous functions on Mrówka’s spaces, Proc. Am. Math. Soc., 133, 2137-2146 (2005) · Zbl 1085.46015
[26] Koszmider, P.; Laustsen, N. J., A Banach space induced by an almost disjoint family, admitting only few operators and decompositions · Zbl 07319242
[27] Kuratowski, K., Topology, vol. 1 (1966), Academic Press and PWN · Zbl 0158.40901
[28] Marciszewski, W., A function space \(C(K)\) not weakly homeomorphic to \(C(K) \times C(K)\), Stud. Math., 88, 2, 129-137 (1988) · Zbl 0666.46022
[29] Marciszewski, W., On a classification of pointwise compact set of the first Baire class functions, Fundam. Math., 133, 195-2009 (1989) · Zbl 0719.54022
[30] Marciszewski, W., Rosenthal compacta, (Encyclopedia of General Topology (2003), Elsevier), 142-144
[31] Marciszewski, W.; Plebanek, G., Extension operators and twisted sums of \(c_0\) and \(C(K)\) spaces, J. Funct. Anal., 274, 1491-1529 (2018) · Zbl 1390.46016
[32] Marciszewski, W.; Pol, R., On Banach spaces whose norm-open sets are \(F_\sigma\) sets in the weak topology, J. Math. Anal. Appl., 350, 708-722 (2009) · Zbl 1167.46311
[33] Marciszewski, W.; Pol, R., On Borel almost disjoint families, Monatshefte Math., 168, 545-562 (2012) · Zbl 1298.54022
[34] Mardešic, S., On covering dimension and inverse limits of compact spaces, Ill. J. Math., 4, 2, 278-291 (1960) · Zbl 0094.16902
[35] Mercourakis, S., Some remarks on countably determined measures and uniform distribution of sequences, Monatshefte Math., 121, 79-111 (1996) · Zbl 0901.28009
[36] Okunev, O., The Lindelöf number of \(C_p(X) \times C_p(X)\) for strongly zero-dimensional X, Cent. Eur. J. Math., 9, 978-983 (2011) · Zbl 1245.54018
[37] Semadeni, Z., Banach spaces non-isomorphic to their Cartesian squares. II, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., 8, 2, 81-86 (1960) · Zbl 0091.27802
[38] Soukup, L., Scattered spaces, (Encyclopedia of General Topology (2003), Elsevier), 350-353
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.