arXiv subjects

Daisuke Hirota

Publications and source records attributed to Daisuke Hirota.

10 recordsLinked to original sources

arXiv2026

Surjective isometries on the positive parts of the unit spheres of some function spaces

We consider the space $C^1[0, 1]$ of continuously differentiable functions on the closed unit interval $[0, 1]$ and the space $\operatorname{Lip}[0, 1]$ of Lipschitz continuous functions on $[0, 1]$, equipped with the norms \begin{align*} \|f\|_{\sigma, p} = \begin{cases} \sqrt[p]{|f(0)|^p + \|f'\|_\infty^p} & (1 \le p < \infty), \\ \max\{\, |f(0)|, \|f'\|_\infty \,\} & (p = \infty). \end{cases} \end{align*} We show that every surjective isometry on the positive part of the unit sphere extends to a surjective complex-linear isometry on the entire space. As a corollary, every such isometry also extends to an isometric order isomorphism on the real subspaces $C^1_{\mathbb{R}}[0, 1]$ and $\operatorname{Lip}_{\mathbb{R}}[0, 1]$.

math.FA↗

arXiv2026

On the Fischer-Musz\'ely equation for the positive cones of $C^*$-algebras

We study the Fischer-Musz\'ely functional equation for the positive semidefinite and the positive definite cones of unital $C^*$-algebras. We show that any bijection between the positive semidefinite cones satisfying the Fischer-Musz\'ely equality extends to a Jordan $*$-isomorphism followed by multiplication on both sides by a positive element. As a corollary, we obtain a similar result for the positive definite cones of unital $C^*$-algebras.

math.FA↗

arXiv2026

Globalization of local sign structures for phase-isometries on uniform algebras

We study surjective phase-isometries between the unit spheres of uniform algebras. Although such maps preserve maximal convex sets up to signs, the resulting local sign ambiguity prevents a direct application of the usual Banach--Stone type arguments for isometries. The main point of the paper is to prove that these local sign structures can be globalized on the Choquet boundary. To this end, we refine an additive Bishop-type construction and use it to propagate the sign information among the maximal convex sets associated with boundary points. As a consequence, every surjective phase-isometry admits a boundary representation by means of a global sign function, a unimodular weight, a homeomorphism between the Choquet boundaries, and a clopen decomposition into complex-linear and conjugate-linear parts. We then extend this representation to the maximal ideal spaces and obtain the corresponding real-algebraic Banach--Stone type representation.

math.FA↗

arXiv2026

The Impossibility of Cohesion Without Fragmentation

Accounts of social division and cohesion presuppose relations whose maintenance already depends on positional compatibility. We develop this feasibility layer as a static theory: a positioning event fixes players' positions, and a compatibility function determines which pairs can sustain a relation. Two results follow from one formalism. First, any non-uniform compatibility pattern on the dyads sharing a positional requirement forces fragmentation and cohesion together; the only events that guarantee no collapse are those rendering every pair compatible. The asymmetry between unilateral severance and bilateral confirmation recovers the convention underlying pairwise stability without a behavioral premise, while a separate specialization, in which compatibility partitions the position space, yields assortative survival without choice or opportunity. Second, at the measure level, the survival rate factorizes exactly into integration and cohesion terms. Along any complete relaxation path that relaxes one pair at a time (the generic case under distance thresholds with absolutely continuous positions), the cohesion term is non-monotone once at least three players are present, and in clustered configurations it is bounded above by a sharp asymptotic value that finite systems approach but never attain, with strict crossing of the two terms once the number of clusters exceeds two.

physics.soc-ph↗

arXiv2025

Norm additive mappings between commutative $C^{*}$-algebras in the range

Let $ A_i $ be a commutative $ C^{*} $-algebra for $ i = 1, 2 $, and denote by $ A_i^{+} $ its positive cone, consisting of all positive elements of $ A_i $. In this paper, we investigate surjective, not necessarily continuous mappings $ T: A_1^{+} \to A_2^{+} $ that satisfy the norm equality \[ \| T(a + b) \| = \| T(a) + T(b) \| \quad (a, b \in A_1^{+}). \] We prove that such a mapping $ T $ is necessarily additive and positive homogeneous. Furthermore, we show that if the mapping $T:A_{1}^{+}\to A_{2}^{+}$ between the positive cones of two unital commutative $C^{*}$-algebras $A_{i}$ with the unit element $ 1_{A_i} $ for $ i = 1, 2 $, and if $ T $ is also injective, then $T(1_{A_1})^{-1}T$ is a composition operator. This is the submitted version of a paper currently under minor revision for the Journal of Mathematical Analysis and Applications.

math.FA↗

arXiv2025

Rational Adversaries and the Maintenance of Fragility: A Game-Theoretic Theory of Rational Stagnation

Cooperative systems often remain in persistently suboptimal yet stable states. This paper explains such "rational stagnation" as an equilibrium sustained by a rational adversary whose utility follows the principle of potential loss, $u_{D} = U_{ideal} - U_{actual}$. Starting from the Prisoner's Dilemma, we show that the transformation $u_{i}' = a\,u_{i} + b\,u_{j}$ and the ratio of mutual recognition $w = b/a$ generate a fragile cooperation band $[w_{\min},\,w_{\max}]$ where both (C,C) and (D,D) are equilibria. Extending to a dynamic model with stochastic cooperative payoffs $R_{t}$ and intervention costs $(C_{c},\,C_{m})$, a Bellman-style analysis yields three strategic regimes: immediate destruction, rational stagnation, and intervention abandonment. The appendix further generalizes the utility to a reference-dependent nonlinear form and proves its stability under reference shifts, ensuring robustness of the framework. Applications to social-media algorithms and political trust illustrate how adversarial rationality can deliberately preserve fragility.

cs.GT↗

arXiv2024

Phase-isometries between the positive cones of the Banach space of continuous real-valued functions

For a locally compact Hausdorff space $L$, we denote by $C_0(L,\mathbb{R})$ the Banach space of all continuous real-valued functions on $L$ vanishing at infinity equipped with the supremum norm. We prove that every surjective phase-isometry $T\colon C_0^+(X,\mathbb{R}) \to C_0^+(Y,\mathbb{R})$ between the positive cones of $C_0(X,\mathbb{R})$ and $C_0(Y,\mathbb{R})$ is a composition operator induced by a homeomorphism between $X$ and $Y$. Furthermore, we show that any surjective phase-isometry $T\colon C_0^+(X,\mathbb{R}) \to C_0^+(Y,\mathbb{R})$ extends to a surjective linear isometry from $C_0(X,\mathbb{R})$ onto $C_0(Y,\mathbb{R})$.

math.FA↗

arXiv2022

Every commutative JB$^*$-triple satisfies the complex Mazur--Ulam property

We prove that every commutative JB$^*$-triple satisfies the complex Mazur--Ulam property. Thanks to the representation theory, we can identify commutative JB$^*$-triples as spaces of complex-valued continuous functions on a principal $\mathbb{T}$-bundle $L$ in the form $$C_0^\mathbb{T}(L):=\{a\in C_0(L):a(λt)=λa(t)\text{ for every } (λ,t)\in\mathbb{T}\times L\}.$$ We prove that every surjective isometry from the unit sphere of $C_0^\mathbb{T}(L)$ onto the unit sphere of any complex Banach space admits an extension to a surjective real linear isometry between the spaces.

math.FA↗

arXiv2021

Exploring new solutions to Tingley's problem for function algebras

In this note we present two new positive answers to Tingley's problem in certain subspaces of function algebras. In the first result we prove that every surjective isometry between the unit spheres, $S(A)$ and $S(B)$, of two uniformly closed function algebras $A$ and $B$ on locally compact Hausdorff spaces can be extended to a surjective real linear isometry from $A$ onto $B$. In a second goal we study surjective isometries between the unit spheres of two abelian JB$^*$-triples represented as spaces of continuous functions of the form $$C^{\mathbb{T}}_0 (X) := \{ a \in C_0(X) : a (λt) = λa(t) \hbox{ for every } (λ, t) \in \mathbb{T}\times X\},$$ where $X$ is a (locally compact Hausdorff) principal $\mathbb{T}$-bundle. We establish that every surjective isometry $Δ: S(C_0^{\mathbb{T}}(X))\to S(C_0^{\mathbb{T}}(Y))$ admits an extension to a surjective real linear isometry between these two abelian JB$^*$-triples.

math.FA↗

arXiv2021

Surjective isometries between unitary sets of unital JB$^*$-algebras

This paper is, in a first stage, devoted to establish a topological--algebraic characterization of the principal component, $\mathcal{U}^0 (M)$, of the set of unitary elements, $\mathcal{U} (M)$, in a unital JB$^*$-algebra $M$. We arrive to the conclusion that, as in the case of unital C$^*$-algebras, $$\begin{aligned}\mathcal{U}^0(M) &= M^{-1}_{\textbf{1}}\cap\mathcal{U} (M) =\left\lbrace U_{e^{i h_n}}\cdots U_{e^{i h_1}}(\textbf{1}) \colon \begin{array}{c} n\in \mathbb{N}, \ h_j\in M_{sa} \forall\ 1\leq j \leq n \end{array} \right\rbrace \end{aligned}$$ is analytically arcwise connected. Our second goal is to provide a complete description of the surjective isometries between the principal components of two unital JB$^*$-algebras $M$ and $N$. Contrary to the case of unital C$^*$-algebras, we shall deduce the existence of connected components in $\mathcal{U} (M)$ which are not isometric as metric spaces. We shall also establish necessary and sufficient conditions to guarantee that a surjective isometry $Δ: \mathcal{U}(M)\to \mathcal{U} (N)$ admits an extension to a surjective linear isometry between $M$ and $N$, a conclusion which is not always true. Among the consequences it is proved that $M$ and $N$ are Jordan $^*$-isomorphic if, and only if, their principal components are isometric as metric spaces if, and only if, there exists a surjective isometry $Δ: \mathcal{U}(M)\to \mathcal{U}(N)$ mapping the unit of $M$ to an element in $\mathcal{U}^0(N)$. These results provide an extension to the setting of unital JB$^*$-algebras of the results obtained by O. Hatori for unital C$^*$-algebras.

math.OA↗