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A note on elementary equivalence of C(K) spaces

Published online by Cambridge University Press:  12 March 2014

S. Heinrich ,
C. Ward Henson  and
L. C. Moore Jr.
S. Heinrich
Affiliation:
Institute of Mathematics, Academy of Sciences, 1086 Berlin, GDR
C. Ward Henson
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
L. C. Moore Jr.
Affiliation:
Department of Mathematics, Duke University, Durham, North Carolina 27706
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Extract

In this paper we give a closer analysis of the elementary properties of the Banach spaces C(K), where K is a totally disconnected, compact Hausdorff space, in terms of the Boolean algebra B(K) of clopen subsets of K. In particular we sharpen a result in [4] by showing that if B(K1) and B(K2) satisfy the same sentences with ≤ n alternations of quantifiers, then the same is true of C(K1) and C(K2). As a consequence we show that for each n there exist C(K) spaces which are elementarily equivalent for sentences with ≤ n quantifier alternations, but which are not elementary equivalent in the full sense. Thus the elementary properties of Banach spaces cannot be determined by looking at sentences with a bounded number of quantifier alternations.

The notion of elementary equivalence for Banach spaces which is studied here was introduced by the second author [4] and is expressed using the language of positive bounded formulas in a first-order language for Banach spaces. As was shown in [4], two Banach spaces are elementarily equivalent in this sense if and only if they have isometrically isomorphic Banach space ultrapowers (or, equivalently, isometrically isomorphic nonstandard hulls.)

We consider Banach spaces over the field of real numbers. If X is such a space, Bx will denote the closed unit ball of X, Bx = {x ϵ X∣ ∣∣x∣∣ ≤ 1}. Given a compact Hausdorff space K, we let C(K) denote the Banach space of all continuous real-valued functions on K, under the supremum norm. We will especially be concerned with such spaces when K is a totally disconnected compact Hausdorff space. In that case B(K) will denote the Boolean algebra of all clopen subsets of K. We adopt the standard notation from model theory and Banach space theory.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 2 , June 1987 , pp. 368 - 373
Copyright
Copyright © Association for Symbolic Logic 1987

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References

REFERENCES

[1]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[2]Eršov, Yu., Decidability of the elementary theory of relatively complemented distributive lattices and the theory of filters, Algebra i Logika, vol. 3 (1964), no 3, pp. 17–38. (in Russian)Google Scholar
[3]Heinrich, S., Henson, C. W. and Moore, J. C. Jr., Elementary equivalence of L1-preduals, Banach space theory and its applications, proceedings, Bucharest, 1981, Lecture Notes in Mathematics, vol. 991, Springer-Verlag, Berlin, 1983, pp. 79–90.Google Scholar
[4]Henson, C. W., Nonstandard hulls of Banach spaces, Israel Journal of Mathematics, vol. 25 (1976), pp. 108–144.CrossRefGoogle Scholar
[5]Henson, C. W., Banach space model theory. I (to appear).Google Scholar
[6]Henson, C. W. and Moore, L. C. Jr., The Banach spaces lp(n) for large p and n, Manuscripta Mathematica, vol. 44 (1983), pp. 1–33.CrossRefGoogle Scholar
[7]Tarski, A., Arithmetical classes and types of Boolean algebras, Bulletin of the American Mathematical Society, vol. 55 (1949), p. 64.Google Scholar