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Isomorphic classification of projective tensor products of spaces of continuous functions

R. M. Causey, E. Galego, C. Samuel

TL;DR

The authors provide a complete isomorphic classification of projective tensor products C(K) ⊗π C(L) for infinite countable K,L, showing each such product is isomorphic to exactly one member of the family C(ω^{ω^ξ}) ⊗π C(ω^{ω^ζ}) with ζ ≤ ξ < ω1, and that no C(M) space can arise as such a tensor product. Central to the approach is the development of two ordinal-indexed invariants, S_ξ and G_{ξ,ζ}, built from Schreier families and the Szlenk index, which refine the Szlenk-based analysis and distinguish tensor products that Szlenk alone cannot. The paper establishes lifting lemmas for weakly null sequences, develops a calculus of weighted Schreier blocks (via p^ξ and q^{ξ,ζ}), and uses intricate Cantor-scheme constructions and Grothendieck-type bounds to bound and lower-bound these invariants. Together, these results extend classical classification results for C(K) spaces to their projective tensor products, with explicit isomorphism-classes determined by Cantor-Bendixson indices and ordinal invariants, and provide precise criteria for when a tensor product can or cannot be isomorphic to a C(M).

Abstract

We prove that for infinite, countable, compact, Hausdorff spaces $K,L$, $C(K)\widehat{\otimes}_πC(L)$ is isomorphic to exactly one of the spaces $C(ω^{ω^ξ})\widehat{\otimes}_πC(ω^{ω^ζ})$, $0\leqslant ζ\leqslant ξ<ω_1$. We also prove that $C(K)\widehat{\otimes}_πC(L)$ is not isomorphic to $C(M)$ for any compact, Hausdorff space $M$.

Isomorphic classification of projective tensor products of spaces of continuous functions

TL;DR

The authors provide a complete isomorphic classification of projective tensor products C(K) ⊗π C(L) for infinite countable K,L, showing each such product is isomorphic to exactly one member of the family C(ω^{ω^ξ}) ⊗π C(ω^{ω^ζ}) with ζ ≤ ξ < ω1, and that no C(M) space can arise as such a tensor product. Central to the approach is the development of two ordinal-indexed invariants, S_ξ and G_{ξ,ζ}, built from Schreier families and the Szlenk index, which refine the Szlenk-based analysis and distinguish tensor products that Szlenk alone cannot. The paper establishes lifting lemmas for weakly null sequences, develops a calculus of weighted Schreier blocks (via p^ξ and q^{ξ,ζ}), and uses intricate Cantor-scheme constructions and Grothendieck-type bounds to bound and lower-bound these invariants. Together, these results extend classical classification results for C(K) spaces to their projective tensor products, with explicit isomorphism-classes determined by Cantor-Bendixson indices and ordinal invariants, and provide precise criteria for when a tensor product can or cannot be isomorphic to a C(M).

Abstract

We prove that for infinite, countable, compact, Hausdorff spaces , is isomorphic to exactly one of the spaces , . We also prove that is not isomorphic to for any compact, Hausdorff space .
Paper Structure (24 sections, 42 theorems, 268 equations)

This paper contains 24 sections, 42 theorems, 268 equations.

Table of Contents

  1. Introduction
  2. Tensor products and $p$-weakly summable sequences
  3. On Cantor-Bendixson index of compact Hausdorff spaces
  4. On Cantor scheme on compact Hausdorff spaces
  5. On trees
  6. On Schreier families $\mathcal{S}_\xi$ and $\mathcal{S}_\zeta[\mathcal{S}_\xi].$
  7. On Szlenk index $Sz(X)$
  8. On families of probability measures on $\mathcal{S}_\xi$
  9. Some convenient weights on Schreier families
  10. The weight $p^\xi$ on $\mathcal{S}_\xi$
  11. The weight $q^{\xi,\zeta}$ on $\mathcal{S}_\zeta[\mathcal{S}_\xi]$
  12. On weakly null sequences and weakly null collections in quotient spaces
  13. The properties ${S}_\xi$ and ${G}_{\xi,\zeta}$ for weakly null collections in Banach spaces
  14. On operators having $\gamma$-$S_\xi$ and $\gamma$-$G_{\xi, \zeta}$ properties
  15. On classes of operators $\mathbf{S}_\xi$ and $\mathbf{G}_{\xi,\zeta}$ and the functionals $\mathbf{s}_\xi$ and $\mathbf{g}_{\xi,\zeta}$
  16. ...and 9 more sections

Key Result

Theorem 1.1

If $K$ and $L$ are infinite countable compact metric spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is isomorphic to exactly one of the spaces $C(\omega^{\omega^\xi})\widehat{\otimes}_\pi C(\omega^{\omega^\zeta})$, $\zeta\leqslant \xi<\omega_1$.

Theorems & Definitions (92)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Lemma 6.1
  • Remark 6.2
  • Lemma 6.3
  • ...and 82 more