A complete classification of the spaces of compact operators on $C([1,\alpha ], l_{p})$ spaces, $1<p< \infty$
HTML articles powered by AMS MathViewer
- by Dale E. Alspach and Elói Medina Galego PDF
- Proc. Amer. Math. Soc. 143 (2015), 2495-2506 Request permission
Abstract:
We complete the classification, up to isomorphism, of the spaces of compact operators on $C([1, \gamma ], l_{p})$ spaces, $1<p< \infty$. In order to do this, we classify, up to isomorphism, the spaces of compact operators ${\mathcal K}(E, F)$, where $E= C([1, \lambda ], l_{p})$ and $F=C([1, \xi ], l_q)$ for arbitrary ordinals $\lambda$ and $\xi$ and $1< p \leq q< \infty$.
More precisely, we prove that it is relatively consistent with ZFC that for any infinite ordinals $\lambda$, $\mu$, $\xi$ and $\eta$ the following statements are equivalent:
-
[(a)] ${\mathcal K}(C([1, \lambda ], l_{p}), C([1, \xi ], l_{q}))$ is isomorphic to ${\mathcal K}(C([1, \mu ], l_{p}), C([1, \eta ], l_{q})) .$
-
[(b)] $\lambda$ and $\mu$ have the same cardinality and $C([1,\xi ])$ is isomorphic to $C([1, \eta ])$ or there exists an uncountable regular ordinal $\alpha$ and $1 \leq m, n < \omega$ such that $C([1, \xi ])$ is isomorphic to $C([1, \alpha m])$ and $C([1, \eta ])$ is isomorphic to $C([1, \alpha n])$.
Moreover, in ZFC, if $\lambda$ and $\mu$ are finite ordinals and $\xi$ and $\eta$ are infinite ordinals, then the statements (a) and (b$’$) are equivalent.
-
[(b$’$)] $C([1,\xi ])$ is isomorphic to $C([1, \eta ])$ or there exists an uncountable regular ordinal $\alpha$ and $1 \leq m, n \le \omega$ such that $C([1, \xi ])$ is isomorphic to $C([1, \alpha m])$ and $C([1, \eta ])$ is isomorphic to $C([1, \alpha n])$.
References
- Dale E. Alspach and Elói Medina Galego, Geometry of the Banach spaces $C(\beta \Bbb N\times K,X)$ for compact metric spaces $K$, Studia Math. 207 (2011), no. 2, 153–180. MR 2864387, DOI 10.4064/sm207-2-4
- C. Bessaga and A. Pełczyński, Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions, Studia Math. 19 (1960), 53–62. MR 113132, DOI 10.4064/sm-19-1-53-62
- Andreas Defant and Klaus Floret, Tensor norms and operator ideals, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. MR 1209438
- J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
- G. A. Edgar, Measurability in a Banach space. II, Indiana Univ. Math. J. 28 (1979), no. 4, 559–579. MR 542944, DOI 10.1512/iumj.1979.28.28039
- R. J. Gardner and W. F. Pfeffer, Borel measures, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 961–1043. MR 776641
- Elói Medina Galego, On subspaces and quotients of Banach spaces $C(K,X)$, Monatsh. Math. 136 (2002), no. 2, 87–97. MR 1914222, DOI 10.1007/s006050200036
- Elói Medina Galego, An isomorphic classification of $C({\bf 2}^{\mathfrak {m}}\times [0,\alpha ])$ spaces, Bull. Pol. Acad. Sci. Math. 57 (2009), no. 3-4, 279–287. MR 2581093, DOI 10.4064/ba57-3-9
- Elói Medina Galego, On isomorphic classifications of spaces of compact operators, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3335–3342. MR 2515403, DOI 10.1090/S0002-9939-09-09828-1
- Elói Medina Galego, Complete isomorphic classifications of some spaces of compact operators, Proc. Amer. Math. Soc. 138 (2010), no. 2, 725–736. MR 2557189, DOI 10.1090/S0002-9939-09-10117-X
- Elói Medina Galego, On spaces of compact operators on $C(K,X)$ spaces, Proc. Amer. Math. Soc. 139 (2011), no. 4, 1383–1386. MR 2748430, DOI 10.1090/S0002-9939-2010-10544-0
- S. P. Gul′ko and A. V. Os′kin, Isomorphic classification of spaces of continuous functions on totally ordered bicompacta, Funkcional. Anal. i Priložen. 9 (1975), no. 1, 61–62 (Russian). MR 0377489
- S. Heĭnrih, Approximation properties in tensor products, Mat. Zametki 17 (1975), 459–466 (Russian). MR 412835
- Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
- William B. Johnson and Joram Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 1–84. MR 1863689, DOI 10.1016/S1874-5849(01)80003-6
- A. Kanamori and M. Magidor, The evolution of large cardinal axioms in set theory, Higher set theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977) Lecture Notes in Math., vol. 669, Springer, Berlin, 1978, pp. 99–275. MR 520190
- Thomas Kappeler, Banach spaces with the condition of Mazur, Math. Z. 191 (1986), no. 4, 623–631. MR 832820, DOI 10.1007/BF01162352
- S. V. Kisljakov, A classification of the spaces of continuous functions on the ordinals, Sibirsk. Mat. Ž. 16 (1975), 293–300, 420 (Russian). MR 0377490
- Denny H. Leung, On Banach spaces with Mazur’s property, Glasgow Math. J. 33 (1991), no. 1, 51–54. MR 1089953, DOI 10.1017/S0017089500008028
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
- Haskell P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36. MR 270122, DOI 10.4064/sm-37-1-13-36
- C. Samuel, Sur la reproductibilité des espaces $l_{p}$, Math. Scand. 45 (1979), no. 1, 103–117 (French). MR 567436, DOI 10.7146/math.scand.a-11828
- C. Samuel, Sur les sous-espaces de $l_{p}\hat \otimes l_{q}$, Math. Scand. 47 (1980), no. 2, 247–250 (French). MR 612698, DOI 10.7146/math.scand.a-11887
- Albert Wilansky, Mazur spaces, Internat. J. Math. Math. Sci. 4 (1981), no. 1, 39–53. MR 606656, DOI 10.1155/S0161171281000021
Additional Information
- Dale E. Alspach
- Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
- Email: alspach@math.okstate.edu
- Elói Medina Galego
- Affiliation: Department of Mathematics, University of São Paulo, São Paulo, Brazil 05508-090
- MR Author ID: 647154
- Email: eloi@ime.usp.br
- Received by editor(s): August 12, 2013
- Received by editor(s) in revised form: January 1, 2014, and January 8, 2014
- Published electronically: February 3, 2015
- Communicated by: Thomas Schlumprecht
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 2495-2506
- MSC (2010): Primary 46B03; Secondary 46B25
- DOI: https://doi.org/10.1090/S0002-9939-2015-12441-0
- MathSciNet review: 3326031