Borel complexity of isometry classes of $\mathcal{C}(K)$ spaces with countable compacta

Abstract: For every countable compact space $K$, we determine the exact Borel complexity of the isometry class of the Banach space $\mathcal{C}(K)$. As a byproduct, we also determine the precise Borel complexity of the homeomorphism class of a fixed countable compact space $K$, improving earlier results of Cenzer and Mauldin. The above results provide concrete and natural examples of sets with arbitrarily high, still exactly determined, Borel complexity. Moreover, we find a new characterization of those real $L_1$-preduals that are isometric to $\mathcal{C}(K)$ for some zero-dimensional compact space $K$ and we determine the precise Borel complexity of $\mathcal{C}(2<sup>{\mathbb{N}})$.

• The paper provides a precise calculation of the Borel complexity for isometry classes of C(K) spaces, establishing completeness results for various countable compact forms.
• It introduces a geometric norming criterion that characterizes L₁-preduals isometric to C(K) spaces with zero-dimensional K, linking Banach space geometry to descriptive set theory.
• The study unifies topological and metric classification by matching the complexities of homeomorphism and isometry classes, while addressing open questions on summable Szlenk index.

Borel Complexity of Isometry Classes of $\mathcal{C}(K)$ Spaces with Countable Compacta

Introduction and Context

This work establishes the precise descriptive set-theoretic (Borel) complexity of the isometry classes of separable Banach spaces of the form $\mathcal{C}(K)$, where $K$ is a countable compact space. The authors leverage the deep interplay between Banach space theory, descriptive set theory (DST), and the structural theory of compact (specifically, zero-dimensional and countable) spaces. They provide new characterizations of when a real Banach space is isometric to a $\mathcal{C}(K)$ space with $K$ zero-dimensional, as well as analyze the relation between topological and metric (isometric) classification problems. The main technical innovation is the explicit calculation of the Borel complexity of isometry classes for all such $\mathcal{C}(K)$ spaces.

This work advances earlier results concerning the complexity of isomorphism and isometry classes in Banach space theory (notably those using invariant descriptive set theory protocols) and also sharpens classical findings of Cenzer and Mauldin regarding homeomorphism classes of compact metric spaces.

Main Results

Precise Borel Complexities

The core theorem determines the Borel complexity of isometry classes of all $\mathcal{C}(K)$ spaces with $K$ countable compact (or the Cantor set):

• For $K$ homeomorphic to $[0,\omega^{\beta+n}]$ (where $\mathcal{C}(K)$0 is 0 or a countable limit ordinal and $\mathcal{C}(K)$1), the isometry class is $\mathcal{C}(K)$2-complete.
• For $\mathcal{C}(K)$3 homeomorphic to $\mathcal{C}(K)$4 with $\mathcal{C}(K)$5, the isometry class is $\mathcal{C}(K)$6-complete.
• The isometry class of $\mathcal{C}(K)$7 (the Cantor set) is $\mathcal{C}(K)$8-complete.

These results are shown to be optimal, meaning that the classes are not Borel of any lower complexity.

Additionally, the precise Borel complexity of the homeomorphism classes of the underlying compacta in the space of compact subsets of a Polish space is obtained, and in almost all cases matches the complexity of the isometry class of the corresponding $\mathcal{C}(K)$9.

Characterization of Isometric $K$0-Preduals

A central new analytical tool is a geometric condition that characterizes those real $K$1-preduals that are isometric to a $K$2 space with $K$3 zero-dimensional compact. Specifically, the following norming criterion is established (see condition (1) in the paper):

$K$4.

This gives a structurally accessible property (not purely in terms of extreme points) that allows identification of $K$5 spaces in terms of the Banach space geometry alone, independent of the underlying compact.

Analytical/Topological Advances

The paper establishes that (contrary to previous expectations) the class of Banach spaces with summable Szlenk index is $K$6-hard, answering a question posed in (Chen et al., 2023) negatively.

It also demonstrates a tight correspondence between the Borel complexity of topological classification (homeomorphism type of the compacta) and metric classification (isometry class of $K$7), except for the Cantor set, where the isometry class is strictly more complicated than the homeomorphism class.

Technical Approach

The authors adopt and refine the coding of separable Banach spaces via norm functions on a countable $K$8-linear vector space, as developed in their earlier work (Comandur, 2022), equipping the space of Banach spaces with a Polish topology. This enables the precise formulation and calculation of Borel classes and completeness.

To establish upper bounds, new effective characterizations of the relevant isometry classes are derived (notably for $K$9 with $\mathcal{C}(K)$0 countable or the Cantor set), using tools from functional analysis (Szlenk derivatives, properties of $\mathcal{C}(K)$1-preduals, etc). The lower bounds are proved via explicit continuous reductions from standard $\mathcal{C}(K)$2-complete sets, and are built recursively using inductive construction on the Cantor-Bendixson rank, with the one-point compactification technique to handle successor ordinals.

The complexity theory is connected with classical DST via homeomorphism invariants (Mazurkiewicz-Sierpiński and Brouwer classification), and Borel hierarchies are tracked in detail throughout all reductions.

Implications

For Descriptive Set Theory and Banach Space Theory

These results provide the first explicit natural examples in analysis and topology of sets with arbitrarily high, but exactly determined, Borel complexity strictly beyond the first few classes. This directly answers longstanding questions (see e.g. [Kechris 1995, p. 189]) for "naturally arising" highly complex Borel sets.

The explicit characterization of isometry classes for countable $\mathcal{C}(K)$3 provides a template for future classification results for larger or more complex metric compacta, and may help calibrate classification invariants in invariant descriptive set theory.

The geometric criterion for $\mathcal{C}(K)$4-preduals is likely to be useful beyond the present context, in $\mathcal{C}(K)$5-geometry, the theory of $\mathcal{C}(K)$6-algebras, and the study of isometric structure in functional analysis.

For Complexity of Property Classes

The negative resolution of the Borelness of the class of spaces with summable Szlenk index restricts approaches for effective (Borel) classification in asymptotic geometric analysis, and informs the structure of classification problems.

Aggressive Comparison to Homeomorphism Classes

A striking aspect is the comparison between the complexity of homeomorphism classes of compact metric spaces and metric classification. The main result demonstrates that, for almost all countable compacta, the Borel complexity of the isometry class of $\mathcal{C}(K)$7 coincides exactly with that of the homeomorphism class of $\mathcal{C}(K)$8 in a suitable hyperspace, but in the case of the Cantor set the latter is $\mathcal{C}(K)$9, strictly lower than the $K$0-completeness of the isometry class. This explicitly exhibits cases where geometric and topological invariants diverge in descriptive complexity.

Future Directions

Possible extensions include:

• Beyond Countable Compacta: Extension to uncountable scattered compacta, or to non-zero-dimensional $K$1.
• Other Equivalence Relations: Analysis of complexity for almost isometric or Lipschitz equivalence classes.
• Noncommutative Context: Transfer of these techniques to operator systems, noncommutative $K$2-spaces, or $K$3-algebraic contexts.
• Effective Descriptive Set Theory: Investigating effective Borel structures (in the sense of recursion theory) on the isometry classes.

The explicit, constructive nature of the methods also opens avenues for practical algorithms implementing these classification criteria within automated reasoning systems in functional analysis.

Conclusion

This paper offers a systematic and technically rigorous treatment of the Borel complexity of isometry and homeomorphism classes associated to $K$4 spaces with $K$5 countable compacta. It provides optimal bounds and effective characterizations, enriching both the structure theory of Banach spaces and the landscape of natural examples in descriptive set theory. The results and techniques developed have broad implications for the theoretical understanding of classification problems at the interface of topology, measure, and functional analysis.

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References:

• (Comandur, 2022) Marek Cúth et al., "Polish spaces of Banach spaces"
• [CDDK2] Marek Cúth et al., "Polish spaces of Banach spaces: complexity of isometry and isomorphism classes"
• [Kechrisbook] A. S. Kechris, "Classical descriptive set theory"
• [CM82] D. Cenzer, R. D. Mauldin, "On the Borel class of the derived set operator"
• [CM83] D. Cenzer, R. D. Mauldin, "On the Borel class of the derived set operator. II"