arXiv subjects
Publications and source records attributed to Antonio Acuaviva.
We construct two complemented subspaces of $L_1[0,1]$. The first has the Schur property but fails the Radon--Nikodým property. The second contains a copy of $\ell_2$ but no copy of $L_1[0,1]$. Neither space is isomorphic to a Banach lattice. This gives a negative answer to the complemented-subspace question of Lindenstrauss and Rosenthal. A Lean 4 formalisation of the main results accompanies the paper.
We give a negative solution to the complemented subspace problem for Banach spaces with unconditional bases over both the real and complex fields. For every $ρ>0$, we construct a projection $P_ρ$ of norm less than $1+ρ$ on a separable superreflexive space \begin{equation*} X_ρ=\left(\bigoplus_{j=1}^{\infty}\ell_{p_j}^{N_j}\right)_2, \qquad p_j\downarrow2, \end{equation*} such that $Z_ρ=P_ρ(X_ρ)$ and its dual $Z_ρ^*$ have Schauder bases but admit no unconditional bases. Over the real field, both spaces have Gordon--Lewis local unconditional structure (GL-lust) but fail Dubinsky--Pełczyński--Rosenthal local unconditional structure (DPR-lust), disproving a conjecture of Figiel, Johnson and Tzafriri. In particular, neither is isomorphic to a Banach lattice, giving a negative solution to the separable Banach-lattice complemented subspace problem. A modification of the construction also shows that the class of separable real Banach lattices is not primary. A Lean 4 formalisation of the main results accompanies the paper.
We complete Banach's isometric conjecture over the complex field. More precisely, if \(X\) is a complex normed space and, for some \(2\leqslant n<\dim_{\C}X\), all its \(n\)-dimensional complex subspaces are isometric as metric spaces, then the norm is induced by a Hermitian inner product. We also prove the quaternionic counterpart. The central geometric argument first treats real star bodies without convexity or central symmetry; applied to circled complex or quaternionic bodies, it shows that mutually real-linearly equivalent hyperplane sections force the ambient body to be a Hermitian ellipsoid. The proof adapts the bundle-degree mechanism introduced by Lu and Yang for the real case. Finally, we obtain extensions to absolutely homogeneous functions, graded Fréchet spaces, metrisable locally convex spaces, and compatible translation-invariant metrics.
Existing source-seeking algorithms for robot swarms typically require either direct gradient measurements or rigid geometric formations, limiting their flexibility and resilience to robot failures. We propose a fully distributed solution that overcomes these limitations by computing an ascending direction through local field measurements and distributed estimation of centroid-relative coordinates. The resulting architecture consists of three exponentially convergent algorithms operating in a slow-fast closed-loop system, enabling simultaneous estimation and motion control without central coordination. Our framework accommodates arbitrary swarm geometries and analyzes how the spatial distribution of robots affects gradient observability, robustness, and resilience to failures. We characterize optimal swarm shapes that guarantee alignment with the true gradient and show how shape morphing can maneuver the collective motion. The approach is developed for kinematic points in $\mathbb{R}^m$ and extended to 2D unicycles with constant speed. Simulations with large-scale swarms validate the methodology.
Given $A\in\operatorname{GL}(N,2)$ and an integer $K$, we ask whether $A$ can be implemented by at most $K$ CNOT gates on fixed labelled wires with all-to-all connectivity. We prove that this problem is NP-complete. From a finite simple graph $G=(V,E)$, we construct an upper-unitriangular matrix $A_G\in\operatorname{GL}(2|V|+|E|+1,2)$ satisfying $\ell_{\mathrm{CNOT}}(A_G)=2|V|+2|E|+τ(G)$, where $τ(G)$ is the minimum vertex-cover size. Each target matrix has $O(N)$ nonzero entries and row Hamming weight at most four. The lower bound unfolds an arbitrary CNOT circuit into an XOR directed acyclic graph and applies projection--contraction operations, allowing cancellation and unrestricted reuse of intermediate parities. For this family, the optimum is unchanged by any finite number of clean or borrowed ancillary wires that must be restored. A polynomial-time decoder further yields NP-hardness of approximation within every fixed additive constant and, through an L-reduction from Minimum Vertex Cover on cubic graphs, APX-hardness of the associated CNOT-circuit optimisation problem.
We investigate the capacity of current language models to contribute to mathematical research. In Banach space theory, AI systems generated key ideas and proofs for five new results, which were then verified and refined by humans. We also developed an automated system that searches the literature for open problems and attempts solutions at scale. Our results show both the potential of language models for mathematical discovery and the continuing importance of expert verification.
Let $K$ be an uncountable compact metrizable space, and $E$ be a separable Banach space with the uniform primary factorisation property (UPFP) and containing no isomorphic copy of $c_0$. We prove that $C(K, E)$ also has the UPFP. As a consequence, we obtain the primariness of the bi-parameter spaces $C([0,1],\ell_p)$ and $C([0,1], L_p)$ for $1 \leq p<\infty$.
Let $\mathscr{A}(X)$ and $\mathscr{K}(X)$ denote the ideals of approximable and compact operators on a Banach space $X$, respectively. We construct a unital Banach algebra $A$ of density character $\mathfrak c=2^{\aleph_0}$, with exactly one non-zero proper closed two-sided ideal, such that, for every Banach space $X$, the algebra $A$ is isomorphic to neither $\mathscr{B}(X)/\mathscr{A}(X)$ nor $\mathscr{B}(X)/\mathscr{K}(X)$. Thus $A$ is not a Calkin algebra under either of the two customary conventions.
We construct a separable Banach space $X$ with a Schauder basis such that $B_X$ is not a uniformly continuous retract of $B_{X^{**}}$. Consequently, $X$ is not a uniformly continuous retract of $X^{**}$ and hence is not a Lipschitz retract of $X^{**}$.
We show that there is no real or complex indecomposable Banach space with the primary factorisation property (PFP). We relate the PFP of a Banach space $E$ to ring-theoretic infiniteness of $\mathcal{B}(E)$ and of $\mathcal{B}(E)/\mathcal{M}_E$, where $\mathcal{M}_E$ denotes the set of operators not factoring the identity on $E$, in the case it is the unique maximal ideal of $\mathcal{B}(E)$. For complex $E$ with the PFP, this quotient is purely infinite exactly when it is not scalar. We isolate the quantitative gap relevant to ultrapowers, identify classical sequence spaces as positive non-scalar cases, and show that Read's space $E_{\operatorname{R}}$ does not have the uniform PFP.
We prove transfer principles for the uniform primary factorisation property (UPFP) from a Banach space $X$ to the vector-valued sequence spaces $\ell_1(X)$, $c_0(X)$ and $\ell_\infty(X)$. The hypotheses are either finite-cotype assumptions on $X$ or $X^*$, or natural self-similarity assumptions on $X$. Consequently, under these conditions, the resulting vector-valued sequence spaces are primary. As applications, we recover the primariness of $\ell_\infty(L_p)$ for $1\leq p<\infty$ without using Bourgain's localisation method, and obtain the primariness of $c_0(L_1)$. We also show that $\ell_1(Γ,L_1[0,1])$ has the UPFP for every set $Γ$, and consequently that $C[0,1]^*$ has the UPFP and is primary.
We prove that the spaces $\ell_p(C(α))$ and $\ell_p(C[0,1])$ have the uniform primary factorisation property whenever $α$ is an ordinal and $1<p\leq\infty$. For the case $p=1$, we establish a general criterion ensuring that $\ell_1(X)$ inherits the uniform primary factorisation property from $X$. As a consequence, $\ell_p(C(K))$ is primary for every compact metrizable space $K$ and every $1 \leq p \leq \infty$.
We study the relation between primariness of Banach spaces and the stronger operator-theoretic notions of the primary factorisation property (PFP) and the uniform primary factorisation property (UPFP). We revisit several classical primariness arguments and isolate the additional information needed to factor the identity through arbitrary operators. In the separable setting, this recovers quantitative factorisation versions of the Casazza--Kottman--Lin method for spaces with symmetric bases and treats the exceptional cases of $\ell_1$ and $\ell_\infty$. We then develop support-reduction and free-selection tools for uncountable direct sums, allowing one to transfer primariness and the PFP/UPFP from countable building blocks to non-separable $\ell_p$-, $c_0$- and more general symmetric sums. As applications, we obtain, among others, the primariness of $C[0,1]^*$ under the negation of the Continuum Hypothesis and UPFP results for uncountable sums of ordinal $C(α)$-spaces. Finally, using the finite-block representation of $\mathcal B(\ell_p)$, we prove a uniform primary factorisation theorem for the Banach space $\mathcal B(\ell_p)$, $1<p<\infty$, and end with open problems concerning the gap between primariness and factorisation.
We study the representation of non-weakly compact operators between $AL$-spaces. In this setting, we show that every operator admits a best approximant in the ideal of weakly compact operators. Using duality arguments, we extend this result to operators between $C(L)$-spaces where $L$ is extremally disconnected. We also characterize the weak essential norm for operators between $AL$-spaces in terms of factorizations of the identity on $\ell_1$. As a consequence, we deduce that the weak Calkin algebra $\mathscr{B}(E)/\mathscr{W}(E)$ admits a unique algebra norm for every $AL$-space $E$. By duality, similar results are obtained for $C(K)$-spaces. In particular, we prove that for operators $T: L_{\infty}[0,1] \to L_{\infty}[0,1]$ the weak essential norm, the residuum norm, and the De Blasi measure of weak compactness coincide, answering a question of González, Saksman and Tylli.
Extending a method developed by Koszmider and Laustsen for constructing $C(K)$-spaces we produce families of $C(K)$-spaces with few operators relative to a partially ordered set $\mathcal{P}$. Using these spaces, we construct new $C(K)$-spaces whose closed operator ideals can be completely classified. Additionally, we use these spaces to resolve some questions regarding automatic continuity.
We classify the complemented subspaces of $C_0(L \times L)$, where $L$ is an exotic locally compact Hausdorff space recently constructed by Candido under Ostaszewski's $\clubsuit$-principle.
We give a characterization of the operators on the injective tensor product $E \hat{\otimes}_\varepsilon X$ for any separable Banach space $E$ and any (non-separable) Banach space $X$ with few operators, in the sense that any operator $T: X \to X$ takes the form $T = λI + S$ for a scalar $λ\in \mathbb{K}$ and an operator $S$ with separable range. This is used to give a classification of the complemented subspaces and closed operator ideals of spaces of the form $C_0(ω\times K_\mathcal{A})$, where $K_\mathcal{A}$ is a locally compact Hausdorff space induced by an almost disjoint family $\mathcal{A}$ such that $C_0(K_\mathcal{A})$ has few operators.
For a scattered, locally compact Hausdorff space $K$, we prove that the essential norm on the Calkin algebra \break $\mathscr{B}(C_0(K))/\mathscr{K}(C_0(K))$ is a minimal algebra norm. The proof relies on establishing a quantitative factorization for the identity operator on $c_0$ through non-compact operators $T: C_0(K) \to X$, where $X$ is any Banach space that does not contain a copy of $\ell_1$ or whose dual unit ball is weak$^*$ sequentially compact. It follows that, for every ordinal $α$, the algebras $\mathscr{B}(C[0,α]))$ and $\mathscr{B}(C[0,α]))/\mathscr{K}(C[0,α]))$ have an unique algebra norm.