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Elementary equivalence of Cσ(K) spaces for totally disconnected, compact Hausdorff K

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 of the GDR, 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

This paper is a continuation of the authors' paper [7]; in particular, we give a sharper and more useful criterion for the approximate elementary equivalence of Cσ(K) spaces, where K is a totally disconnected compact Hausdorff space. (See Theorem 2 below.) As an application, we obtain a complete description of the Banach spaces X which are approximately equivalent to c0. Namely, X ≡ Ac0 iff X = Cσ(K) where K is a totally disconnected compact Hausdorff space which has a dense set of isolated points and σ is an involutory homeomorphism on K which has a unique fixed point t, and t is not an isolated point. (See Theorem 7.)

These results are derived from an analysis which we give of the elementary theories of structures (B, σ), where B is a Boolean algebra and σ is an involutory automorphism of B which leaves at most one nontrivial ultrafilter invariant. Let U(σ) denote this ultrafilter, if it exists; let U(σ) = B in case σ leaves no nontrivial ultrafilter invariant. We show that the elementary theory of (B, σ) is completely determined by the theory of (B, U(σ)) (and conversely, because U(σ) is definable in (B, σ)). This makes possible the use of the explicit invariants given by Éršov [4] for structures (B, U) where U is an ultrafilter on B. (These generalize the Tarski invariants for Boolean algebras [15].) We also use the Éršov invariants in the proof of our main result.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 1 , March 1986 , pp. 135 - 146
Copyright
Copyright © Association for Symbolic Logic 1986

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References

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