One-Sided Positive Banach–Mazur Distance

• The one-sided positive Banach–Mazur distance is a refined metric that measures the minimal distortion required by enforcing positivity constraints in isomorphisms between C(K) spaces.
• It highlights the asymmetry in embeddings by linking operator norms with topological invariants like Cantor–Bendixson height and providing explicit quantitative bounds.
• Methodologies include decomposing spaces into clopen slices and optimizing weight factors, thereby resolving long-standing questions regarding integer versus non-integral distance values.

The one-sided positive Banach–Mazur distance is a refinement of the classical Banach–Mazur distance that introduces positivity constraints into the context of isomorphisms between Banach spaces of continuous functions on compact spaces. This metric quantifies the minimal distortion required to realize a one-sided positive isomorphism, elucidating the interplay between linear structure, order properties, and topological invariants such as Cantor–Bendixson height. Recent research has established both qualitative and quantitative results for this distance, identifying critical thresholds and demonstrating the non-integrality of certain canonical cases, thereby resolving longstanding questions in the theory of $C(K)$ spaces (Cúth et al., 16 Jan 2026).

1. Definition of the One-Sided Positive Banach–Mazur Distance

For compact spaces $K$ and $L$, let $C(K)$ and $C(L)$ denote the Banach spaces of real-valued continuous functions endowed with the supremum norm. The classical Banach–Mazur distance is defined as

$$d_{BM}(C(K), C(L)) := \inf\left\{ \|T\| \|T^{-1}\| : T \colon C(K) \xrightarrow{\simeq} C(L) \right\},$$

where the infimum is over all surjective linear isomorphisms $T$.

The one-sided positive Banach–Mazur distance introduces the following positivity constraint: a bounded linear operator $T: C(K)\rightarrow C(L)$ is called a one-sided positive isomorphism if $T$ is surjective and $T(f) \ge 0$ for all $f \ge 0$, but $T^{-1}$ need not be positive. The distance is then

$$d^+_{BM}(C(K), C(L)) := \inf\left\{ \|T\| \|T^{-1}\| : T \text{ is a one-sided positive isomorphism } C(K) \to C(L) \right\}.$$

This variant is inherently asymmetric, i.e., $d^+_{BM}(C(K), C(L))$ need not equal $d^+_{BM}(C(L), C(K))$.

2. Comparison with the Classical Banach–Mazur Distance

The classical Banach–Mazur distance considers general isomorphisms with no positivity requirements. In contrast, $d^+_{BM}$ restricts to operators preserving the nonnegative cone. This imposes additional constraints, yielding the following fundamental relation: $d_{BM}(C(K), C(L)) \le d^+_{BM}(C(K), C(L)),$ with the possibility of strict and asymmetric inequality.

A critical distinction arises in the field of Cantor–Bendixson height: if $\mathrm{height}(K) > \mathrm{height}(L)$, $d^+_{BM}(C(K), C(L)) = +\infty$, whereas $d_{BM}(C(K), C(L)) < \infty$ provided $C(K) \simeq C(L)$. This highlights the coupling between positivity-preserving embeddings and topological properties of the underlying compacta.

3. Key Classification and Quantitative Theorems

Two principal results underpin the structure of $d^+_{BM}$ for $C(K)$ spaces on infinite countable compacta:

Qualitative Classification (Theorem A): For infinite countable compacta $K, L$,

• There exists a one-sided positive isomorphism $T: C(K) \xrightarrow{\simeq} C(L)$ if and only if $C(K) \simeq C(L)$ and $\mathrm{height}(K) \le \mathrm{height}(L)$.

For arbitrary compact $K, L$, the existence of a positive embedding $C(K) \hookrightarrow C(L)$ implies $\mathrm{height}(K) \le \mathrm{height}(L)$.

Quantitative Estimates (Theorem B): For countable ordinal $\alpha > 0$ and $n \in \{2,3,\ldots\}$:

• $$d_{BM}\left((\omega^{\alpha}), (\omega^{\alpha}\,n)\right) \le d^+_{BM}\left((\omega^{\alpha}), (\omega^{\alpha}\,n)\right) \le 2+\sqrt{5}$$,
• $$2n-1 \le d^+_{BM}\left((\omega^{\alpha} n), (\omega^{\alpha})\right) \le n + \sqrt{(n-1)(n+3)}$$,
• If $\alpha = \omega^{\beta}$, then

$$d_{BM}\left((\omega^{\alpha}), (\omega^{\alpha n})\right) = d^+_{BM}\left((\omega^{\alpha}), (\omega^{\alpha n})\right) = n + \sqrt{(n-1)(n+3)}.$$

4. Proof Sketch of the Exact Formula

The proof that

$$d_{BM}\left(C(\omega^{\omega^\alpha}), C(\omega^{\omega^\alpha n})\right) = n + \sqrt{(n-1)(n+3)}$$

proceeds by establishing matching upper and lower bounds.

Upper Bound Construction: $[1, \omega^{\omega^\alpha n}]$ is decomposed into $n$ clopen slices $I_1, \ldots, I_n$, each homeomorphic to $[1, \omega^{\omega^\alpha}]$. An operator $T$ is formed by aggregating the functions on each slice with weights $\lambda_i$ in the simplex $\sum_i \lambda_i=1$. The operator $$T: C(\omega^{\omega^\alpha}) \to C(\omega^{\omega^\alpha n})$$ is positive, and the product $\|T\| \|T^{-1}\|$ is optimized over $\lambda_i$, yielding

$C(n) = n + \sqrt{(n-1)(n+3)},$

as shown via the infimum

$$C(n) = \inf_{\lambda \in \Delta_n} \max\left\{ \frac{2}{\lambda_1}-1, \frac{2}{\lambda_2}+1, \ldots, \frac{2}{\lambda_n}+1 \right\}.$$

Lower Bound via Height Arguments: Assuming $\|T\| \|T^{-1}\| < n + \sqrt{(n-1)(n+3)}$, one constructs $n+1$ disjointly supported test functions whose images cannot fit into a compact of height $\omega^\alpha$, leading to a contradiction by sup-norm estimates and properties of the triangle inequality. For $\beta = \omega^\alpha$, this involves explicit construction using Proposition 2.4 of [rondos–somaglia] and a “pigeonhole”-and-signing argument, enforcing the lower bound.

5. Representative Cases and Applications

Case ($\alpha$)	Domain/Target	Value / Bound for $d^+_{BM}$
$\alpha=1$	$C(\omega)$, $C(\omega n)$	$\leq 2+\sqrt{5}$
$\alpha=1$	$C(\omega n)$, $C(\omega)$	$\geq 2n-1$
$\alpha=\omega$	$C(\omega^\omega)$, $C(\omega^{\omega n})$	$= n + \sqrt{(n-1)(n+3)}$

In the balanced case $C(\omega) \to C(\omega^n)$, the distance matches the exact value $n + \sqrt{(n-1)(n+3)}$. For $\alpha = \omega$, the exact positive and classical Banach–Mazur distances coincide.

These explicit non-integer values conclusively resolve a 50-year-old question by Bessaga–Pełczyński–Pelczyński concerning the integrality of Banach–Mazur distances between countable $C(K)$ spaces. The one-sided positive variant provides directional information: showing in which embedding directions positivity can be achieved, and quantifying the minimal distortion required.

6. Interplay with Cantor–Bendixson Height and Structural Implications

The Cantor–Bendixson height is pivotal in understanding one-sided positivity. The existence of a positive embedding $C(K)\hookrightarrow C(L)$ necessitates $\mathrm{height}(K)\le \mathrm{height}(L)$ for scattered compacta. Theorems and propositions within (Cúth et al., 16 Jan 2026) and prior work by rondos–somaglia delineate the propagation and limitation of height under positive maps, forming the backbone for asymmetric estimates in $d^+_{BM}$.

This coupling of functional, order-theoretic, and topological invariants delivers new insights into classification problems for $C(K)$ spaces beyond the symmetric algebraic setting. A plausible implication is that positivity constraints naturally encode geometric complexity otherwise invisible to purely isomorphic classification.

7. Research Significance and Future Directions

The introduction and exact evaluation of the one-sided positive Banach–Mazur distance provide definitive answers to longstanding classification problems in the theory of Banach spaces of continuous functions. These results expand the landscape of possible distances, proving the existence of non-integral values and highlighting the asymmetric and topologically sensitive nature of positivity-preserving isomorphisms.

Taken together, the frameworks and estimations in (Cúth et al., 16 Jan 2026) enrich the theory of $C(K)$ spaces and illuminate the role of order structure in Banach space theory. Future work may aim to generalize these results to non-countable compacta, seek sharper bounds for other classes of function spaces, or explore applications in the structure theory for ordered Banach spaces.