Positive Linear Functionals Research Articles

Generalizations of the Kantorovich and Wielandt Inequalities with Applications to Statistics

By utilizing the properties of positive definite matrices, mathematical expectations, and positive linear functionals in matrix space, the Kantorovich inequality and Wielandt inequality for positive definite matrices and random variables are obtained. Some novel Kantorovich type inequalities pertaining to matrix ordinary products, Hadamard products, and mathematical expectations of random variables are provided. Furthermore, several interesting unified and generalized forms of the Wielandt inequality for positive definite matrices are also studied. These derived inequalities are then exploited to establish an inequality regarding various correlation coefficients and study some applications in the relative efficiency of parameter estimation of linear statistical models.

On the truncated matricial moment problem. I

This paper is about the general truncated matrix-valued moment problem. Let Hq denote the complex Hermitian q×q-matrices, q∈N. Suppose that (X,X) is a measurable space and E is a finite-dimensional vector space of measurable mappings of X into Hq. A linear functional Λ on E is called a moment functional if there exists a positive Hq-valued measure μ on (X,X) such that Λ(F)=∫X〈F,dμ〉 for F∈E.We prove a matricial version of the Richter–Tchakaloff theorem which states that each moment functional on E has a finitely atomic representing measure. It is shown that strictly positive linear functionals on E are moment functionals. For a moment functional Λ, we study the set of atoms W(Λ) and define two Carathéodory numbers of Λ. Further, we define and investigate the core set V(Λ) which generalizes the core variety from the scalar case to the matricial setting. A main result of the paper is the equality W(Λ)=V(Λ).

Exotic C⁎-algebras of geometric groups

We consider a new class of potentially exotic group C*-algebras C⁎(PFp⁎(G)) for a locally compact group G, and its connection with the class of potentially exotic group C*-algebras CLp⁎(G) introduced by Brown and Guentner. Surprisingly, these two classes of C*-algebras are intimately related. By exploiting this connection, we show CLp⁎(G)=C⁎(PFp⁎(G)) for p∈(2,∞), and the C*-algebras CLp⁎(G) are pairwise distinct for p∈(2,∞) when G belongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of CLp⁎(G) and C⁎(PFp⁎(G)). This greatly generalizes earlier results of Okayasu (see [30]) and the second author (see [40]) on the pairwise distinctness of CLp⁎(G) for 2<p<∞ when G is either a noncommutative free group or the group SL(2,R), respectively.As a byproduct of our techniques, we present two applications to the theory of unitary representations of a locally compact group G. Firstly, we give a short proof of the well-known Cowling-Haagerup-Howe Theorem, which presents sufficient condition implying the weak containment of a cyclic unitary representation of G in the left regular representation of G (see [14]). Also we give a near solution to a 1978 conjecture of Cowling stated in [10]. This conjecture of Cowling states if G is a Kunze-Stein group and π is a unitary representation of G with cyclic vector ξ such that the mapG∋s↦〈π(s)ξ,ξ〉 belongs to Lp(G) for some 2<p<∞, then Aπ⊆Lp(G). We show Bπ⊆Lp+ϵ(G) for every ϵ>0 (recall Aπ⊆Bπ).

Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction

Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide a similar construction in the case of real Jordan algebras. Given a real, finite-dimensional, formally real Jordan algebra ${\mathcal J}$, we exploit the generalized distribution determined by the Jordan product on the dual ${\mathcal J}^{\star}$ to induce a pseudo-Riemannian metric tensor on the leaves of the distribution. In particular, these leaves are the orbits of a Lie group, which is the structure group of ${\mathcal J}$, in clear analogy with what happens for coadjoint orbits. However, this time in contrast with the Lie-algebraic case, we prove that not all points in ${\mathcal J}^{*}$ lie on a leaf of the canonical Jordan distribution. When the leaves are contained in the cone of positive linear functionals on ${\mathcal J}$, the pseudo-Riemannian structure becomes Riemannian and, for appropriate choices of ${\mathcal J}$, it coincides with the Fisher-Rao metric on non-normalized probability distributions on a finite sample space, or with the Bures-Helstrom metric for non-normalized, faithful quantum states of a finite-level quantum system, thus showing a direct link between the mathematics of Jordan algebras and both classical and quantum information geometry.

Gleason’s theorem for composite systems

Gleason’s theorem (Gleason 1957 J. Math. Mech. 6 885) is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that finitely additive measures on the projection lattice extend to positive linear functionals on the algebra of bounded operators . Over many years, and by the effort of various authors, the theorem has been broadened in its scope from type I to arbitrary von Neumann algebras (without type factors). Here, we prove a generalisation of Gleason’s theorem to composite systems. To this end, we strengthen the original result in two ways: first, we extend its scope to dilations in the sense of Naimark (1943 Dokl. Akad. Sci. SSSR 41 359) and Stinespring (1955 Proc. Am. Math. Soc. 6 211) and second, we require consistency with respect to dynamical correspondences on the respective (local) algebras in the composition (Alfsen and Shultz 1998 Commun. Math. Phys. 194 87). We show that neither of these conditions changes the result in the single system case, yet both are necessary to obtain a generalisation to bipartite systems.

Symmetry Reduction of States I

We develop a general theory of symmetry reduction of states on (possibly non-commutative) *-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra g . The key idea advocated for in this article is that the "correct" notion of positivity on a *-algebra A is not necessarily the algebraic one whose positive elements are the sums of Hermitian squares a^* a with a \in A , but can be a more general one that depends on the example at hand, like pointwise positivity on *-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on A thus depends on this choice of positivity on A , and the notion of positivity on the reduced algebra A_{red} should be such that states on A_{red} are obtained as reductions of certain states on A . We discuss three examples in detail: Reduction of the *-algebra of smooth functions on a Poisson manifold M , which reproduces the coisotropic reduction of M ; reduction of the Weyl algebra with respect to translation symmetry; and reduction of the polynomial algebra with respect to a U(1)-action.

Parametric models and information geometry on W*-algebras

We introduce the notion of smooth parametric model of normal positive linear functionals on possibly infinite-dimensional W^{star }-algebras generalizing the notions of parametric models used in classical and quantum information geometry. We then use the Jordan product naturally available in this context in order to define a Riemannian metric tensor on parametric models satsfying suitable regularity conditions. This Riemannian metric tensor reduces to the Fisher–Rao metric tensor, or to the Fubini-Study metric tensor, or to the Bures–Helstrom metric tensor when suitable choices for the W^{star }-algebra and the models are made.

Strict Quantization of Polynomial Poisson Structures

We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on {mathbb {R}}^d, generalizing known results for constant and linear Poisson structures to polynomial Poisson structures of arbitrary degree. We give several examples of nonlinear Poisson structures and construct explicit formal star products whose deformation parameter can be evaluated to any real value of hbar , giving strict quantizations on the space of analytic functions on {mathbb {R}}^d with infinite radius of convergence. We also address further questions such as continuity of the classical limit hbar rightarrow 0, compatibility with ^*-involutions, and the existence of positive linear functionals. The latter can be used to realize the strict quantizations as ^*-algebras of operators on a pre-Hilbert space which we demonstrate in a concrete example.

Three term relations for multivariate Uvarov orthogonal polynomials

Three term relations for orthogonal polynomials in several variables associated to a moment linear functional obtained by a Uvarov modification of a given moment functional are studied. Existence of Uvarov orthogonal polynomials is analyzed, stating conditions to ensure it. The matrices of the three term relations of the Uvarov orthogonal polynomials are explicitly given in terms of the matrices of the three term relations satisfied by the original family. Two examples are presented in order to show that the results are valid for positive definite linear functionals and also for some quasi definite linear functionals which are not positive definite.

Dirichlet polynomials and a moment problem

Consider a linear functional L defined on the space \({\mathcal {D}}[s]\) of Dirichlet polynomials with real coefficients and the set \({\mathcal {D}}_+[s]\) of non-negative elements in \({\mathcal {D}}[s].\) An analogue of the Riesz–Haviland theorem in this context asks: What are all \({\mathcal {D}}_+[s]\)-positive linear functionals L, which are moment functionals? Since the space \({\mathcal {D}}[s],\) when considered as a subspace of \(C([0, \infty ), {\mathbb {R}}),\) fails to be an adapted space in the sense of Choquet, the general form of Riesz–Haviland theorem is not applicable in this situation. In an attempt to answer the forgoing question, we arrive at the notion of a moment sequence, which we call the Hausdorff log-moment sequence. Apart from an analogue of the Riesz–Haviland theorem, we show that any Hausdorff log-moment sequence is a linear combination of \(\{1, 0, \ldots , \}\) and \(\{f(\log (n)\}_{n \geqslant 1}\) for a completely monotone function \(f : [0, \infty ) \rightarrow [0, \infty ).\) Moreover, such an f is uniquely determined by the sequence in question.

Noncommutative rational Clark measures

Abstract We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $\mathbb {C} ^d$ is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers. Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital $C^*$ -algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions.

Gelfand–Naimark theorems for ordered -algebras

Abstract The classical Gelfand–Naimark theorems provide important insight into the structure of general and of commutative $C^*$ -algebras. It is shown that these can be generalized to certain ordered $^*$ -algebras. More precisely, for $\sigma $ -bounded closed ordered $^*$ -algebras, a faithful representation as operators is constructed. Similarly, for commutative such algebras, a faithful representation as complex-valued functions is constructed if an additional necessary regularity condition is fulfilled. These results generalize the Gelfand–Naimark representation theorems to classes of $^*$ -algebras larger than $C^*$ -algebras, and which especially contain $^*$ -algebras of unbounded operators. The key to these representation theorems is a new result for Archimedean ordered vector spaces V: If V is $\sigma $ -bounded, then the order of V is induced by the extremal positive linear functionals on V.

Riesz representation theorems for positive linear operators

We generalise the Riesz representation theorems for positive linear functionals on \(\text {C}_{\text {c}}(X)\) and \(\text {C}_{\text {0}}(X)\), where X is a locally compact Hausdorff space, to positive linear operators from these spaces into a partially ordered vector space E. The representing measures are defined on the Borel \(\sigma \)-algebra of X and take their values in the extended positive cone of E. The corresponding integrals are order integrals. We give explicit formulas for the values of the representing measures at open and at compact subsets of X. Results are included where the space E need not be a vector lattice, nor a normed space. Representing measures exist, for example, for positive linear operators into Banach lattices with order continuous norms, into the regular operators on KB-spaces, into the self-adjoint linear operators on complex Hilbert spaces, and into JBW-algebras.

An Unbounded Generalization of Tomita's Observable Algebras

In this paper we shall build the basic theory of unbounded Tomita's observable algebras which are related to unbounded operator algebras, especially unbounded Tomita-Takesaki theory, operator algebras on Krein spaces, studies of positive linear functionals on *-algebras and so on.

Almost all positive continuous linear functionals can be extended

Let F be an ordered topological vector space (over mathbb {R}) whose positive cone F_+ is weakly closed, and let E subseteq F be a subspace. We prove that the set of positive continuous linear functionals on E that can be extended (positively and continuously) to F is weak-* dense in the topological dual wedge E_+'. Furthermore, we show that this result cannot be generalized to arbitrary positive operators, even in finite-dimensional spaces.

On the convergence of sequences of positive linear operators and functionals on bounded function spaces

Of concern are some simple criteria about the convergence of sequences of positive linear operators and functionals in the framework of spaces of bounded functions which are continuous on a given subset of their domain. Among other things some applications concerning the behaviour of the iterates of Bernstein operators defined both on [ 0 , 1 ] [0,1] and on the d d -dimensional simplex and hypercube ( d ≥ 1 ) (d\geq 1) are discussed. A final section treats the behaviour of integrated arithmetic means with respect to a probability Borel measure on a convex compact subset K K of a locally convex space. As a consequence a general weak law of large numbers for sequences of K − K- valued random variables is derived.

On positive linear functionals and operators associated with generalized means

The paper is concerned with the study of the limit behaviour of the sequences of the positive linear functionals and operators associated with integrated generalized means defined with respect to a given probability Borel measure in the framework of Borel convex subsets of a Hilbert space. The main results are easily achieved through some new Korovkin-type theorems for composition operators and for functionals which are established in the context of function spaces defined on a metric space. Several applications are shown in the special cases of bounded and unbounded real intervals which involve the most common integrated means. Furthermore, some consequences concerning the convergence in distribution, and hence stochastic, of generalized means of vector-valued random variables are also presented. Finally the paper ends with an application related to the so-called box integral problem which refers to the problem to evaluate the limit behaviour as n→∞ of the average distance between two points of [0,1]n randomly chosen according to a given distribution on [0,1]n.

Grüss and Grüss-Voronovskaya-type estimates for complex convolution polynomial operators

A classical well-known result in approximation theory is the Grüss inequality for positive linear functionals, which gives an upper bound for the Chebyshev-type functional. Starting also from a problem posed by Gavrea, this inequality was also investigated in terms of the least concave majorants of the moduli of continuity and for positive linear operators by Acu, Gonska, Rasa and Rusu, where the cases of classical Hermite-Fejer and Fejer-Korovkin convolution operators were considered. Refined versions of the Grüss-type inequality in the spirit of Voronovskaya's theorem were obtained by Gal and Gonska for Bernstein and Paltanea operators of real variable and for complex Bernstein, genuine Bernstein-Durrmeyer and Bernstein-Faber operators attached to analytic functions of complex variable. After the appearance of these papers, several papers by other authors have developed these directions of research. The goal of this paper is to continue the above mentioned directions of research, obtaining Grüss and Grüss-Voronovskaya exact estimates (with respect to the degree of polynomials) for the de la Vallee-Poussin complex polynomials in Section 2, for Zygmund-Riesz complex polynomials in Section 3 and for Jackson complex polynomials in Section 4.

Inequalities of the Edmundson-Lah-Ribarč type for n-convex functions with applications

UDC 517.5 We derive some Edmundson – Lah – Ribarič type inequalities for positive linear functionals and -convex functions. Main results are applied to the generalized -divergence functional. Examples with Zipf – Mandelbrot law are used to illustrate the results.

The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realization

The concept of Gauss quadrature can be generalized to approximate linear functionals with complex moments. Following the existing literature, this survey will revisit such generalization. It is well known that the (classical) Gauss quadrature for positive definite linear functionals is connected with orthogonal polynomials, and with the (Hermitian) Lanczos algorithm. Analogously, the Gauss quadrature for linear functionals is connected with formal orthogonal polynomials, and with the non-Hermitian Lanczos algorithm with look-ahead strategy; moreover, it is related to the minimal partial realization problem. We will review these connections pointing out the relationships between several results established independently in related contexts. Original proofs of the Mismatch Theorem and of the Matching Moment Property are given by using the properties of formal orthogonal polynomials and the Gauss quadrature for linear functionals.