Stone's Theorem Research Articles

Stone's theorem for distributional regression in Wasserstein distance

We extend the celebrated Stone's theorem to the framework of distributional regression. More precisely, we prove that weighted empirical distributions with local probability weights satisfying the conditions of Stone's theorem provide universally consistent estimates of the conditional distributions, where the error is measured by the Wasserstein distance of order p ≥ 1 . Furthermore, for p = 1, we determine the minimax rates of convergence on specific classes of distributions. We finally provide some applications of these results, including the estimation of conditional tail expectation or probability weighted moments.

Linear isometries of noncommutative L0$L_0$‐spaces

AbstractThe description of (commutative and noncommutative) ‐isometries has been studied thoroughly since the seminal work of Banach. In the present paper, we provide a complete description for the limiting case, isometries on noncommutative ‐spaces, which extends the Banach–Stone theorem and Kadison's theorem for isometries of von Neumann algebras. The result is new even in the commutative setting.

Fubini’s theorem for Daniell integrals

We show that in the theory of Daniell integration iterated integrals may always be formed, and the order of integration may always be interchanged. By this means, we discuss product integrals and show that the related Fubini theorem holds in full generality. The results build on a density theorem on Riesz tensor products due to Fremlin, and on the Fubini–Stone theorem.

Banach–Stone theorems for disjointness preserving relations

Banach–Stone theorems for disjointness preserving relations

The strongest Banach–Stone theorem for C0(K,ℓ22)$C_{0}(K, \ell _2^2)$ spaces

AbstractAs usual denote by the real two‐dimensional Hilbert space. We prove that if and are locally compact Hausdorff spaces and is a linear isomorphism from onto satisfying then and are homeomorphic.This theorem is the strongest of all the other vector‐valued Banach–Stone theorems known so far in the sense that in none of them the distortion of the isomorphism , denoted by , is as large as .Some remarks on the proof method developed here to prove our theorem suggest the conjecture that it is in fact very close to the optimal Banach–Stone theorem for spaces, or in more precise words, the exact value of the Banach–Stone constant of is between and .

Banach–Stone theorems for algebras of holomorphic germs on spaces with the approximation property

AbstractIt is shown that if and are balanced compact determining subsets of separable Banach spaces with the approximation property, then the algebras of holomorphic germs and are topologically algebra isomorphic if, and only if, and (the polynomial convex hulls of K and L) are biholomorphically equivalent.

The structure space of C−(X) via that of Γ-semirings

The structure spaces, endowed with the hull kernel topology, of maximal regular congruences, which are prime on a [Formula: see text]-semiring as well as the structure space of prime congruences on a [Formula: see text]-semiring, have been studied via operator semirings. This study has been used to obtain some important results of the structure space of [Formula: see text] of nonpositive real valued continuous functions over a topological space [Formula: see text]. It has been found that the structure spaces of the semiring [Formula: see text] of nonnegative real valued continuous functions and the [Formula: see text]-semiring [Formula: see text] are homeomorphic. Moreover it has been shown that the structure space of [Formula: see text] is the Stone–Čech compactification of [Formula: see text], where [Formula: see text] is a Tychonoff space. Furthermore, the [Formula: see text]-semiring analogue of the ‘Banach–Stone Theorem’ has been obtained.

Uniform Logical New Proofs for the Daniell–Stone Theorem and the Riesz Representation Theorem

Uniform Logical New Proofs for the Daniell–Stone Theorem and the Riesz Representation Theorem

On vertex-induced weighted Turán problems

Recently, Bennett, English and Talanda-Fisher introduced the vertex-induced weighted Turán problem. In this paper, we consider their open Turán problem under sum-edge-weight function and characterize the extremal structure of K ℓ -free graphs. Based on these results, we propose a generalized version of the Erdős–Stone theorem for weighted graphs under two types of vertex-induced weight functions.

A weak vector-valued Banach–Stone theorem for Choquet simplices

A weak vector-valued Banach–Stone theorem for Choquet simplices

The Banach–Stone Theorem in Finsler $$C^*$$-Modules

We study the Banach–Stone theorem in the framework of Hilbert $$C^*$$ -modules and Finsler $$C^*$$ -modules and show that each $$\varphi $$ -morphism T between Hilbert $$C^*$$ -modules and Finsler $$C^*$$ -modules is a weighted composition operator of the form $$Tf(y) = h(y)f(\theta (y))$$ .

Stone type representations and dualities by power set ring

In this paper, it is shown that the Boolean ring of a commutative ring is isomorphic to the ring of clopens of its prime spectrum. In particular, Stone's Representation Theorem is generalized. The prime spectrum of the Boolean ring of a given ring R is identified with the Pierce spectrum of R . The discreteness of prime spectra is characterized. It is also proved that the space of connected components of a compact space X is isomorphic to the prime spectrum of the ring of clopens of X . As another major result, it is shown that a morphism of rings between complete Boolean rings preserves suprema if and only if the induced map between the corresponding prime spectra is an open map.

Quantum phase transitions in nonhermitian harmonic oscillator

The Stone theorem requires that in a physical Hilbert space {{{mathcal {H}}}} the time-evolution of a stable quantum system is unitary if and only if the corresponding Hamiltonian H is self-adjoint. Sometimes, a simpler picture of the evolution may be constructed in a manifestly unphysical Hilbert space {{{mathcal {K}}}} in which H is nonhermitian but {{mathcal {PT}}}-symmetric. In applications, unfortunately, one only rarely succeeds in circumventing the key technical obstacle which lies in the necessary reconstruction of the physical Hilbert space {{{mathcal {H}}}}. For a {{mathcal {PT}}}-symmetric version of the spiked harmonic oscillator we show that in the dynamical regime of the unavoided level crossings such a reconstruction of {{{mathcal {H}}}} becomes feasible and, moreover, obtainable by non-numerical means. The general form of such a reconstruction of {{{mathcal {H}}}} enables one to render every exceptional unavoided-crossing point tractable as a genuine, phenomenologically most appealing quantum-phase-transition instant.

The independence of Stone’s Theorem from the Boolean Prime Ideal Theorem

We give a permutation model in which Stone's Theorem (every metric space is paracompact) is false and the Boolean Prime Ideal Theorem (every ideal in a Boolean algebra extends to a prime ideal) is true. The erring metric space in our model attains only rational distances and is not metacompact. Transfer theorems give the comparable independence in the Zermelo-Fraenkel setting, answering a question of Good, Tree and Watson.

Non-surjective coarse version of the Banach–Stone theorem

Let X and Y be compact Hausdorff spaces and let $$F:C(X)\rightarrow C(Y)$$ be a (non-surjective) coarse $$(M,\varepsilon )$$ -quasi-isometry with $$M<\sqrt{8/7}$$ . Assume also that the range of F “approximates” C(Y) well, then X and Y are homeomorphic. Moreover, if $$U:C(X)\rightarrow C(Y)$$ is the canonical isometry induced by the homeomorphism, then $$\begin{aligned} \Vert F(f)-MU(f)\Vert \le 8\frac{M^2-1}{M}\Vert f\Vert +\varDelta \varepsilon \end{aligned}$$ for every $$f\in C(K)$$ , where $$\varDelta $$ depends only on M and the degree of approximation of C(Y) by the range of F. (See the Introduction for the precise technical approximation condition and the estimate of $$\varDelta $$ ). The approximation result is new even for non-surjective isometries.

Conditioned local limit theorems for random walks defined on finite Markov chains

Let $$(X_n)_{n\geqslant 0}$$ be a Markov chain with values in a finite state space $${\mathbb {X}}$$ starting at $$X_0=x \in {\mathbb {X}}$$ and let f be a real function defined on $${\mathbb {X}}$$. Set $$S_n=\sum _{k=1}^{n} f(X_k)$$, $$n\geqslant 1$$. For any $$y \in {\mathbb {R}}$$ denote by $$\tau _y$$ the first time when $$y+S_n$$ becomes non-positive. We study the asymptotic behaviour of the probability $${\mathbb {P}}_x \left( y+S_{n} \in [z,z+a],\, \tau _y > n \right) $$ as $$n\rightarrow +\infty .$$ We first establish for this probability a conditional version of the local limit theorem of Stone. Then we find for it an asymptotic equivalent of order $$n^{3/2}$$ and give a generalization which is useful in applications. We also describe the asymptotic behaviour of the probability $${\mathbb {P}}_x \left( \tau _y = n \right) $$ as $$n\rightarrow +\infty $$.

Geometry of Spaces of Orthogonally Additive Polynomials on C(K)

We study the space of orthogonally additive n-homogeneous polynomials on C(K). There are two natural norms on this space. First, there is the usual supremum norm of uniform convergence on the closed unit ball. As every orthogonally additive n-homogeneous polynomial is regular with respect to the Banach lattice structure, there is also the regular norm. These norms are equivalent, but have significantly different geometric properties. We characterise the extreme points of the unit ball for both norms, with different results for even and odd degrees. As an application, we prove a Banach–Stone theorem. We conclude with a classification of the exposed points.

A characterisation of isometries with respect to the Lévy-Prokhorov metric

According to the fundamental work of Yu.V. Prokhorov, the general theory of stochastic processes can be regarded as the theory of probability measures in complete separable metric spaces. Since stochastic processes depending upon a continuous parameter are basically probability measures on certain subspaces of the space of all functions of a real variable, a particularly important case of this theory is when the underlying metric space has a linear structure. Prokhorov also provided a concrete metrisation of the topology of weak convergence today known as the L{\'e}vy--Prokhorov distance. Motivated by these facts, the famous Banach--Stone theorem, and some recent works related to characterisations of onto isometries of spaces of Borel probability measures, here we give a complete description of surjective isometries with respect to the L{\'e}vy--Prokhorov metric in case when the underlying metric space is a separable Banach space. Our result can be considered as a generalisation of L. Moln\'ar's earlier Banach--Stone-type result which characterises onto isometries of the space of all probability distribution functions on the real line wit respect to the L{\'e}vy distance. However, the present more general setting requires the development of an essentially new technique.

A solution to the Cambern problem for finite-dimensional Hilbert spaces

Let X be a Hilbert space of real dimension n ≥ 2, and δ > 0 satisfying $$n - 1 < \frac{{2 - \delta }}{{10\delta + 8\sqrt {2\delta + {\delta ^2}} }}.$$ In this paper, it is proven that if K and S are locally compact Hausdorff spaces and T is an isomorphism from C0(K,X) onto C0(S,X) satisfying $$||T||||{T^{ - 1}}|| < \sqrt {2 + \delta }, $$ then K and S are homeomorphic. This solves a long-standing open problem posed by Cambern on Hilbert-valued Banach–Stone theorems via isomorphisms T with distortion ||T|| ||T−1|| strictly greater than $$\sqrt 2 $$ .

The space of probabilistic 1-Lipschitz maps

We introduce and study the natural notion of probabilistic 1-Lipschitz maps. We use the space of all probabilistic 1-Lipschitz maps to give a new method for the construction of probabilistic metric completion (respectively of probabilistic invariant metric group completion). Our construction is of independent interest. We prove that the space of all probabilistic 1-Lipschitz maps defined on a probabilistic invariant metric group can be endowed with a monoid structure. Next, we explicit the set of all invertible elements of this monoid and characterize probabilistic invariant complete Menger groups by the space of all probabilistic 1-Lipschitz maps in the spirit of the classical Banach–Stone theorem.