Complex Banach Conjecture Overview

• Complex Banach Conjecture is a hypothesis identifying Hilbert spaces among complex Banach spaces by examining isometric properties of n-dimensional subspaces.
• It employs a combination of convex geometry, topology, and functional analysis to link local geometric rigidity with global structure.
• Recent progress has resolved several cases using topological reduction and geometric analysis, while open problems continue to stimulate further research.

The Complex Banach Conjecture concerns the characterization of Hilbert spaces among complex Banach spaces based on the local isometric geometry of their $n$-dimensional subspaces. The conjecture posits that if a finite-dimensional complex Banach space $V$ has the property that all its complex $n$-dimensional subspaces are isometric (for some $1 < n < \dim_\mathbb{C} V$), then $V$ itself must be isometric to a Hilbert space. This assertion, formulated by Stefan Banach in 1932, connects the rigidity of subspace geometry with global structure and has driven significant developments at the intersection of geometry, topology, and functional analysis.

1. Statement and Functional-Analytic Formulation

A complex Banach space $(V, \|\cdot\|)$ is a vector space over $\mathbb{C}$ equipped with a norm satisfying the triangle inequality, homogeneity under complex scalars, and completeness with respect to the metric $d(x, y) = \|x - y\|$. An $n$-dimensional subspace $W \subset V$ is called isometric to another $n$-dimensional subspace $W'$ if there exists a surjective $\mathbb{C}$-linear map $T: W \to W'$ with $\|T(x)\| = \|x\|$ for all $x \in W$.

A finite-dimensional complex Banach space is a Hilbert space if and only if its norm arises from an inner product, equivalently, if its closed unit ball $B = \{x \in V: \|x\| \leq 1\}$ is a complex ellipsoid, meaning it is the image of a Euclidean unit ball under an invertible $\mathbb{C}$-linear map.

The conjecture can be equivalently expressed as follows. Let $B \subset \mathbb{C}^{n+1}$ be a compact, convex, $S^1$-invariant set (the unit ball of a complex norm) such that every linear section by an $n$-dimensional complex subspace is linearly equivalent (over $\mathbb{C}$) to every other. Then $B$ must be a complex ellipsoid (Bracho et al., 2020).

2. Historical Progress and Partial Results

The infinite-dimensional real case was settled by Dvoretzky (1960) and the infinite-dimensional complex case by Milman (1971), confirming that under the conjecture’s hypotheses, the space is Hilbertian. In the finite-dimensional real setting, Auerbach–Mazur–Ulam (1935) proved the conjecture for $n=2$.

Gromov (1967) established the conjecture for all even $n$, both real and complex, employing deep codimension arguments; for the complex case, he proved codimension results for codim $> 2n$. Bor–Hernández Lamoneda–Jiménez-Desantiago–Montejano (2019) addressed the remaining real cases with $n \equiv 1 \pmod 4$ (except potentially $n=133$).

The currently open finite-dimensional cases in the complex setting are when $n \equiv 3 \pmod 4$ and $n < \dim V < 2n$, as well as certain exceptional real cases for $n \equiv 3 \pmod 4$ (notably $n=133$ and dimension $134$) (Bracho et al., 2020).

3. Convex Geometric and Topological Methods for $n \equiv 1 \pmod{4}$

For $n \equiv 1 \pmod 4$, Bracho–Montejano deploy a two-step argument:

1. Topological Reduction: The bundle sequence

$$\mathrm{SU}(n) \longrightarrow \mathrm{SU}(n+1) \longrightarrow S^{2n+1}$$

associated with the canonical complex $n$-plane bundle over the sphere is considered. By analyzing the automorphism group $G_K$ preserving a fixed model section $K \subset \mathbb{C}^n$, it is shown that the structure group reduces either to $\mathrm{SU}(n)$ (implying $K$ is the Euclidean ball) or to a subgroup isomorphic to $\mathrm{SU}(n-1)$ (forcing $K$ to be a "complex body of revolution").

2. Convex Geometry of Bodies of Revolution: If every hyperplane section of $B$ is $\mathbb{C}$-linearly equivalent to a fixed revolution body $K$, then $B$ itself must be a complex ellipsoid.

Formally, the main theorem established is:

Theorem (Bracho–Montejano, Theorem 1.1): Let $B \subset \mathbb{C}^{n+1}$, $n \equiv 1 \pmod 4$, $n > 5$, be a compact, convex, $S^1$-invariant body all of whose linear sections by complex $n$-planes are $\mathbb{C}$-linearly equivalent. Then $B$ is a complex ellipsoid (Bracho et al., 2020).

The proof intricately decomposes the reduction of bundle structure groups, convex geometric analysis, and the uniqueness of the ellipsoidal form under these symmetries.

4. The Real Case and the Theory of Complex Structures

For real Banach spaces, a complex structure is defined as a real-linear bounded operator $I$ with $I^2 = -\mathrm{Id}$. The space is denoted $X^I$, with scalar multiplication defined as $(\alpha + i\beta)x := \alpha x + \beta I x$, and two structures $I$ and $J$ are isomorphically equivalent if there exists a real-linear isomorphism $T$ such that $TI = JT$.

The classical form of the complex Banach conjecture for real spaces stated that a separable real Banach space admits either exactly one or else $2^{\aleph_0}$ mutually nonisomorphic complex structures.

Cuellar Carrera constructed Banach spaces with exactly $\omega$ and $\omega_1$ nonisomorphic complex structures, refuting the "one-or-continuum" dogma. Specifically, the existence of a separable reflexive real Banach space $X_\omega$ with exactly countably infinitely many nonisomorphic complex structures, and a (nonseparable) reflexive space $X_{\omega_1}$ with exactly $\omega_1$ nonisomorphic structures, demonstrates that all intermediate cardinalities between finite and continuum can be realized (Cuellar-Carrera, 2014).

5. Technical Constructions and the Geometry of Revolution Bodies

In the setting $n \equiv 1 \pmod 4$, a detailed analysis of the model section $K \subset \mathbb{C}^n$ is essential:

• The group $G_K$ is the subgroup of $\mathrm{SU}(n)$ preserving $K$.
• If $K$ is preserved by all of $\mathrm{SU}(n)$, it must be the Euclidean ball; otherwise, $K$ is a complex body of revolution, meaning there is a distinguished axis $L \subset \mathbb{C}^n$ such that orthogonal hyperplanes to $L$ produce standard $(2n-2)$-balls as sections.
• The geometric analysis employs the stability of complex ellipsoids under their symmetry groups and draws upon the contradiction that arises if $K$ is not itself an ellipsoid, leveraging projection arguments and the structure of bodies of revolution.

The following table summarizes the classification for key values of $n$:

$n$ Modulo 4	Status (Complex Banach)	Main Reference
Even	Settled: $V$ is Hilbert	Gromov (1967) (Bracho et al., 2020)
$n \equiv 1 \pmod 4$	Settled for $n > 5$	Bracho–Montejano (Bracho et al., 2020)
$n \equiv 3 \pmod 4$	Open for $n < \dim V < 2n$	–

6. Remaining Open Problems and Impact

The only unresolved finite-dimensional complex cases are $n \equiv 3 \pmod 4$ with $n < \dim V < 2n$. For real spaces, the isolated case $n=133$ in dimension $134$ remains open.

Cuellar Carrera's results further illuminate the landscape for real Banach spaces: the spectrum of possible cardinalities for nonisomorphic complex structures includes all forms from finite to continuum, shattering the previously held dichotomy and inviting a re-examination of complex structure classification (Cuellar-Carrera, 2014).

Major open questions include the realization of additional cardinalities as the number of nonisomorphic complex structures on a real space, the existence of separable real spaces with exactly $\omega_1$ complex structures, and whether imposing unconditional or symmetric bases enforces uniqueness of complex structure.

7. Connections and Broader Significance

The resolution of the conjecture for $n \equiv 1 \pmod 4$ combines deep topological reduction methods—leveraging the structure of principal $\mathrm{SU}(n)$ bundles—and refined convex-geometric analysis of revolution bodies, bringing the conjecture close to a full solution. These advances signify that the isometric geometry of subspaces imposes strong global constraints, with the Hilbert space structure emerging as the unique solution under rigid subspace isometry conditions. The combinatorial and operator-theoretic constructions in the real case demonstrate an unsuspected richness in possible complex structures, entwining classification problems in Banach space theory with set-theoretic and geometric considerations (Bracho et al., 2020, Cuellar-Carrera, 2014).