Stone's Theorem Research Articles

Noncommutative [formula omitted] structure encodes exactly Jordan structure

We prove that for all 1 ⩽ p ⩽ ∞ , p ≠ 2 , the L p spaces associated to two von Neumann algebras M , N are isometrically isomorphic if and only if M and N are Jordan *-isomorphic. This follows from a noncommutative L p Banach–Stone theorem: a specific decomposition for surjective isometries of noncommutative L p spaces.

A generalization of Bochner-Weil?s theorem and Stone?s theorem on foundation semigroups

In this paper, we improve a result of Youssfi. As a consequence we obtain an extension for the Bochner-Weil theorem and the Stone theorem on a weighted foundation *-semigroup with an identity whose weight is assumed to be bounded on a neighbourhood of the identity.

Quasi-multipliers of operator spaces

We use the injective envelope to study quasi-multipliers of operator spaces. We prove that all representable operator algebra products that an operator space can be endowed with are induced by quasi-multipliers. We obtain generalizations of the Banach–Stone theorem.

Geometric characterization of tripotents in real and complex JB ∗-triples

We establish a geometric characterization of tripotents in real and complex JB ∗-triples. As a consequence we obtain an alternative proof of Kaup's Banach–Stone theorem for JB ∗-triples.

Weighted composition operators and the Gleason–Kahane–Zelazko theorem

The main purpose of this paper is to investigate extensions of the Banach–Stone theorem and of the Holsztynski theorem to some locally multiplicatively convex associative algebras. The proofs are based on a generalization of a theorem, due to A. Gleason, J.-P. Kahane and W. Zelazko, characterizing continuous characters of a unital associative Banach algebra among all its linear forms.

Weighted composition operators and the Gleason?Kahane?Zelazko theorem

The main purpose of this paper is to investigate extensions of the Banach–Stone theorem and of the Holsztynski theorem to some locally multiplicatively convex associative algebras. The proofs are based on a generalization of a theorem, due to A. Gleason, J.-P. Kahane and W. Zelazko, characterizing continuous characters of a unital associative Banach algebra among all its linear forms.

Continuous functions with compact support

The main aim of this paper is to investigate a subring of the ring of continuous functions on a topological space X with values in a linearly ordered field F equipped with its order topology, namely the ring of continuous functions with compact support. Unless X is compact, these rings are commutative rings without unity. However, unlike many other commutative rings without unity, these rings turn out to have some nice properties, essentially in determining the property of X being locally compact non-compact or the property of X being nowhere locally compact. Also, one can associate with these rings a topological space resembling the structure space of a commutative ring with unity, such that the classical Banach Stone Theorem can be generalized to the case when the range field is that of the reals.

Constructing Banaschewski compactification without Dedekind completeness axiom

The main aim of this paper is to provide a construction of the Banaschewski compactification of a zero‐dimensional Hausdorff topological space as a structure space of a ring of ordered field‐valued continuous functions on the space, and thereby exhibit the independence of the construction from any completeness axiom for an ordered field. In the process of describing this construction we have generalized the classical versions of M. H. Stone′s theorem, the Banach‐Stone theorem, and the Gelfand‐Kolmogoroff theorem. The paper is concluded with a conjecture of a split in the class of all zero‐dimensional but not strongly zero‐dimensional Hausdorff topological spaces into three classes that are labeled by inequalities between three compactifications of X, namely, the Stone‐Čech compactification βX, the Banaschewski compactification β0X, and the structure space 𝔐X,F of the lattice‐ordered commutative ring ℭ(X, F) of all continuous functions on X taking values in the ordered field F, equipped with its order topology. Some open problems are also stated.

Local Density in Graphs with Forbidden Subgraphs

A celebrated theorem of Turan asserts that every graph on n vertices with more than $\frac{r\,{-}\,1}{2r}n^2$ edges contains a copy of a complete graph $K_r+1$. In this paper we consider the following more general question. Let G be a $K_r+1-free graph of order n and let α be a constant, 0<α≤1. How dense can every induced subgraph of G on αn vertices be? We prove the following local density extension of Turan's theorem.For every integer $r\geq 2$ there exists a constant $c_r < 1$ such that, if $c_r < \alpha < 1$ and every αn vertices of G span more than\[ \frac{r\,{-}\,1}{2r}(2\alpha -1)n^2\vspace*{7pt} \]edges, then G contains a copy of $K_r+1$. This result is clearly best possible and answers a question of Erdos, Faudree, Rousseau and Schelp [5].In addition, we prove that the only $K_r+1-free graph of order n, in which every αn vertices span at least $\frac{r\,{-}\,1}{2r}(2\alpha -1)n^2$ edges, is a Turan graph. We also obtain the local density version of the Erdos–Stone theorem.

Stone?s representation theorem in fuzzy topology

In this paper, a complete solution to the problem of Stone's repesentation theorem in fuzzy topology is given for a class of completely distributive lattices. Precisely, it is proved that if L is a frame such that 0∈ L is a prime or 1∈ L is a coprime, then the category of distributive lattices is dually equivalent to the category of coherent L -locales and that if L is moreover completely distributive, then the category of distributive lattices is dually equivalent to the category of coherent stratified L -topological spaces.

Positive definite functions on infinite-dimensional convex cones

Part I. Preliminaries and Preparatory Results: Bounded and unbounded operators Cone-valued measures Measures on topological spaces Projective limits of cone-valued measures Holomorphic functions Involutive semigroups and their representations Positive definite kernels and functions $\boldmath{C^*}$-algebras associated with involutive semigroups Integral representations of positive definite functions Convex cones and their faces Examples of convex cones Conelike semigroups: definition and examples Representations of conelike semigroups I Fourier and Laplace transforms Generalized Bochner and Stone Theorems Part II. Main Results: Nussbaum Theorem for open convex cones Positive definite functions on convex cones with non-empty interior Positive definite functions on convex sets Associated Hilbert spaces and representations Nussbaum Theorem for generating convex cones Representations of conelike semigroups II Associated unitary representations Holomorphic extension of unitary representations Holomorphic extension of representations of nuclear groups References Index List of symbols.

Striking differences between ZF and ZF+weak choice in view of metric spaces

Well-known properties held by metric spaces with the countable chain condition (ccc) such as separability, second countability, Lindelöf, paracompactness may fail in the absence of the axiom of choice AC. In the long catalogue of weak choice principles, see [10], we find those which are sufficient (and in some cases necessary) to establish well known equivalences in ZFC between the statements CCMX, where CCMX stands for: ccc metric spaces have the property X, X ∈ {separable, second countable, Lindelöf, paracompact, size ≤ 2 ω }. In addition, we show: (1) The weak form of AC, DC(ℵ1), implies the weaker version of Stone's theorem ccc metric spaces are paracompact and that the latter statement does not imply back DC(ℵ1) in ZF0 (Zermelo-Fraenkel set theory minus the axiom of regularity). (2) In [11] it was shown that Stone's theorem, i.e. metric spaces are paracompact, is deducible from the statement every metric space has a σ-locally finite base and the validity of the reverse implication was in question. We close this gap by showing that the answer to the above question is affirmative.

On some issues related to the moment problem for the band matrices with operator elements

The connection between the classical moment problem and the spectral theory of second order difference operators (or Jacobi matrices) is a thoroughly studied topic. Here we examine a similar connection in the case of the second order operator replaced by an operator generated by an infinite band matrix with operator elements. For such operators, we obtain an analog of the Stone theorem and consider the inverse spectral problem which amounts to restoring the operator from the moment sequence of its Weyl matrix. We establish the solvability criterion for such problems, find the conditions ensuring that the elements of the moment sequence admit an integral representation with respect to an operator valued measure and discuss an algorithm for the recovery of the operator. We also indicate a connection between the inverse problem method and the Hermite–Padé approximations.

Proof of a Conjecture of Bollobás and Kohayakawa on the Erdős–Stone Theorem

For any integer r⩾1, let a(r) be the largest constant a⩾0 such that if ϵ>0 and 0<c<c0 for some small c0=c0(r, ϵ) then every graph G of sufficiently large order n and at least1−1r+cn2 edges contains a copy of any (r+1)-chromatic graph H of independence numberα(H)⩽(a−ϵ)lognlog(1/c). T. Kővári et al. (1954, Colloq. Math.2, 50–57) and B. Bollobás and P. Erdős (1973, Bull. London Math. Soc.5, 317–321) showed that 1⩽a(1)⩽2. In an improvement to the Erdős–Stone theorem (1946), B. Bollobás et al. (1976, J. London Math. Soc. (2)12, 219–224) showed that a(r)>0 for all r and conjectured that liminfr→∞a(r)≠0. V. Chvátal and E. Szemerédi (1981, J. London Math. Soc. (2)23, 207–214) settled it by giving a(r)⩾0.002 for all r. We show that, for all r,a(r)=a(1). Further we prove the conjecture of B. Bollobás and Y. Kohayakawa (1994, Combinatorica14, 279–286). The weak form of it states that for any r⩾1, 0<c<1/r, every graph G of sufficiently large order n⩾n0(r, c) and (1−1/r+c)(n2) edges contains any (r+1)-chromatic graph such that, in a proper vertex coloring, the smallest and the other color classes are of size at leastβlognlog(1/c) and βlognlogr, respectively, for an absolute constant β>0. That is, all color classes but one are relatively large for fixed r, small c→0, and large n→∞. Our proof method is based on Szemerédi's Regularity Lemma.

A Common Extension of the Erdős–Stone Theorem and the Alon–Yuster Theorem for Unbounded Graphs

The Erdős–Stone theorem (1946, Bull. Am. Math. Soc., 52, 1089–1091) and the Alon–Yuster theorem (1992, Graphs Comb., 8, 95–102 ) are both very fundamental in extremal graph theory. We give a common extension of them, which states as follows: For everyϵ> 0 and r≥ 2, there exists c=cϵ, r> 0 such that, for any 0 ≤θ≤ 1, if H is a graph of order | H | ≤clogn and with chromatic number r then every n -vertex graph G with minimum degree at least ( 1 − 1 __ r−1 +θ ____ r(r−1)) n contains at least (θ−ϵ)n/| H | vertex-disjoint copies of H.When θ=ϵr (r− 1) or θ= 1, it would imply the two theorems.The important point is that our theorem enables us to deal with a larger graph H of order | H | →∞(as n→∞), while | H | was fixed in the Alon–Yuster theorem (and in another common extension by Komlós (2000, Combinatorica,20, 203–218)).The bounds clogn and ( 1 − 1 __ r−1 +θ ____r (r−1)) n are both essentially the best possible.

On a Generalization of the Stone–Weierstrass Theorem

A categorical version of the famous theorem of Stone and Weierstrass is formulated and studied in detail. Several applications and examples are given.

New Application of Noncanonical Maps in Quantum Mechanics

One invertible and one unitary operator can be used to reproduce the effect of a q-deformed commutator of annihilation and creation operators. The original annihilation and creation operators are mapped into new operators, not conjugate to each other, whose standard commutator equals the identity plus a correction proportional to the original number operator. The consistency condition for the existence of this new set of operators is derived, by exploiting the Stone theorem on 1-parameter unitary groups. The above scheme leads to modified “equations of motion” which do not preserve the properties of the original first-order set for annihilation and creation operators. Their relation with commutation relations is also studied.

Spectral integration and spectral theory for non‐Archimedean Banach spaces

Banach algebras over arbitrary complete non‐Archimedean fields are considered such that operators may be nonanalytic. There are different types of Banach spaces over non‐Archimedean fields. We have determined the spectrum of some closed commutative subalgebras of the Banach algebra ℒ(E) of the continuous linear operators on a free Banach space E generated by projectors. We investigate the spectral integration of non‐Archimedean Banach algebras. We define a spectral measure and prove several properties. We prove the non‐Archimedean analog of Stone theorem. It also contains the case of C‐algebras C∞(X, 𝕂). We prove a particular case of a representation of a C‐algebra with the help of a ‐projection‐valued measure. We consider spectral theorems for operators and families of commuting linear continuous operators on the non‐Archimedean Banach space.

Results about κ-normality

A regular topological space is called κ- normal if any two disjoint κ-closed subsets in it are separated. In this paper we present some results about κ-normality. In Section 1, the technique of adding isolated points to topological spaces has been used to construct three counterexamples. The first one shows the following statements: (1) A product of two linearly ordered topological spaces need not be κ- normal even if one of the factors is compact. (2) A product of two normal countably compact topological spaces need not be κ- normal. (3) A product of a normal topological space with a compact Hausdorff topological space need not be κ- normal. The second one presents a mad family R⊂[ω] ω such that the Mrówka space Ψ( R) is not κ-normal. The third one will show that a scattered locally compact countably compact topological space need not be κ- normal. In Section 2 we have proved the following κ-normal version of Stone's theorem: If X is κ- normal and countably compact and Y is metrizable, then X× Y is κ- normal. For a κ-normal version of Dowker's theorem, we have been able to prove one direction which is the following statement: If X is not κ- countably metacompact, then X× I is not κ- normal. We use I for the closed unit interval [0,1] with the usual topology. The converse is still unsettled. We will show that if X is a Dowker space, then the Alexandroff Duplicate space A( X) of X is a Dowker space with the property that A( X)× I is not κ- normal. Section 3 has been devoted to the notion of local κ-normality. It will be shown that not every locally κ-normal topological space is κ-normal, even if the space satisfies other topological properties such as locally compactness, metacompactness, or countable compactness.

Inverse asymptotique de la matrice de Toeplitz et noyau de Green

Two generalizations of Spitzer–Stone theorem are given. This theorem (1960) shows a deep link between the asymptotic behaviour of the elements of (T N ( f)) −1, where T N ( f) is the Toeplitz matrix associated to the singular symbol f of order 2, and the Green kernel of the second derivative operator. An exact expression of the inverse is obtained for a particular family, the second result is concerned by an asymptotic expansion for a general family.