Positive Linear Functionals Research Articles

Orthogonality in $$C^{*}$$ C ∗ -algebras

The aim in this paper is to study algebraic orthogonality between positive elements of a $$C^{*}$$ -algebra in the context of geometric orthogonality. It has been shown that the algebraic orthogonality in certain classes of $$C^{*}$$ -algebras is equivalent to geometric orthogonality when supported with some order-theoretic conditions. Further more, algebraic orthogonality between positive elements in a $$C^{*}$$ -algebra is also characterized in terms of positive linear functionals.

English

The aim of the paper is to obtain a very general inequality for convex functions on the triangle. To construct such inequality, we include positive linear functionals on the space of real functions. That inequality can be applied to various forms of mathematical means, and its integral form contains the Jensen, Fej\'{; ; e}; ; r and Hermite-Hadamard inequality.

Finite-dimensional invariant subspace property and amenability for a class of Banach algebras

Motivated by a result of Ky Fan in 1965, we establish a characterization of a left amenable F-algebra (which includes the group algebra and the Fourier algebra of a locally compact group and quantum group algebras, or more generally the predual algebra of a Hopf von Neumann algebra) in terms of a finite-dimensional invariant subspace property. This is done by first revealing a fixed point property for the semigroup of norm one positive linear functionals in the algebra. Our result answers an open question posted in Tokyo in 1993 by the first author. We also show that the left amenability of an ideal in an F-algebra may determine the left amenability of the algebra.

Commutation of Projections and Characterization of Traces on von Neumann Algebras. III

We obtain new necessary and sufficient commutation conditions for nonnegative operators and projections in terms of operator inequalities. It is shown that in the general case in this inequalities the projections cannot be replaced by arbitrary nonnegative operators with preservation of operators commutativity. We also present new necessary and sufficient commutation conditions for projections in terms of operator inequalities. These inequalities are applied for trace characterization on von Neumann algebras in the class of all positive normal functionals. We also consider the following problems: I. Characterization of traces among arbitrary weights on von Neumann algebras. II. Characterization of tracial functionals among all positive linear functionals on C∗-algebras. III. Characterization of commutativity for C∗-algebras.

Classification and GNS-construction for general universal products

It is known that there are exactly five natural products, which are universal products fulfilling two normalization conditions simultaneously. We classify universal products without these extra conditions. We find a two-parameter deformation of the Boolean product, which we call (r, s)-products. Our main result states that, besides degnerate cases, these are the only new universal products. Furthermore, we introduce a GNS-construction for not necessarily positive linear functionals on algebras and study the GNS-construction for (r, s)-product functionals.

Functions Like Convex Functions

The paper deals with convex sets, functions satisfying the global convexity property, and positive linear functionals. Jensen's type inequalities can be obtained by using convex combinations with the common center. Following the idea of the common center, the functional forms of Jensen's inequality are considered in this paper.

Refinements and Generalizations of Functional Dresher’s and Bellman’s Inequalities

We give refinements and generalizations of the Dresher and Bellman inequalities for positive linear functionals. We also give reverse of the new obtained generalized version of these inequalities. Finally, we apply our results on time scales integrals to obtain refinements and generalizations of time scales Dresher’s and Bellman’s inequalities.

Absolute continuity of positive linear functionals

The goal of this paper is to present characterizations for absolute continuity of representable positive functionals on general $^*$-algebras. From the results we give a new and very different proof to our recently published Lebesgue decomposition theorem for representable positive functionals. On unital $C^*$-algebras and measure algebras of compact groups further characterizations are included in the paper. As an application of our results, we answer Gudder's problem on the uniqueness of the Lebesgue decomposition in the case of commutative $^*$-algebras and measure algebras of compact groups. Another application to faithful positive functionals defined on the latter $^*$-algebras is also included.

Application of Localization to the Multivariate Moment Problem

It is explained how the localization technique introduced by the author in [19] leads to a useful reformulation of the multivariate moment problem in terms of extension of positive semidefinite linear functionals to positive semidefinite linear functionals on the localization of $\mathsf{R}[\underline{x}]$ at $p = \prod_{i=1}^n(1+x_i^2)$ or $p' = \prod_{i=1}^{n-1}(1+x_i^2)$. It is explained how this reformulation can be exploited to prove new results concerning existence and uniqueness of the measure $\mu$ and density of $\mathsf{C}[\underline{x}]$ in $\mathscr{L}^s(\mu)$ and, at the same time, to give new proofs of old results of Fuglede [11], Nussbaum [21], Petersen [22] and Schmüdgen [27], results which were proved previously using the theory of strongly commuting self-adjoint operators on Hilbert space.

Extensions of Representable Positive Linear Functionals to Unitized Quasi *-Algebras: A New Method

In this paper we introduce a topological approach for extending a representable linear functional \({\omega}\), defined on a topological quasi *-algebra without unit, to a representable linear functional defined on a quasi *-algebra with unit. In particular, we suppose that \({\omega}\) is continuous and the positive sesquilinear form \({\varphi_\omega}\), associated with \({\omega}\), is closable and prove that the extension \({\overline{\varphi_\omega}^e}\) of the closure \({\overline{\varphi_\omega}}\) is an i.p.s. form. By \({\overline{\varphi_\omega}^e}\) we construct the desired extension.

Free group C*-algebras associated with ℓp

For every p ≥ 2, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to positive linear functionals on the free group C*-algebra associated with the ideal ℓp. This is a generalization of Haagerup's characterization for the case of the reduced free group C*-algebra. As a consequence, the canonical quotient map between the associated C*-algebras is not injective, and they have a unique tracial state.

N-exponential convexity of some dynamic Hardy-type functionals

Recently, some dynamic Hardy-type inequalities with certain kernels are studied in (5) with the help of arbitrary time scales. We use the positive linear functionals obtained from the results of (5) to give non trivial examples of n-exponentially convex functions.

Pseudoquotients on commutative Banach algebras

We consider pseudoquotient extensions of positive linear functionals on a commutative Banach algebra $\mathcal{A}$ and give conditions under which the constructed space of pseudoquotients can be identified with all Radon measures on the structure space $\hat{\mathcal{A}}$.

Čebyšev-Grüss-type inequalities revisited

Abstract We generalize and improve several inequalities of the Čebyšev-Grüss-type using least concave majorants of the moduli of continuity of the functions involved. Our focus is on normalized positive linear functionals. We discuss a problem posed by the two Gavreas and also give the solution of a stronger one. In a section about the non-multiplicativity of positive linear operators it is demonstrated that the previous use of second moments is not quite the right choice. This is documented in the case of the classical Hermite-Fejér and de La Vallée Poussin convolution operators.

On the properties of McShane’s functional and their applications

In this paper we derive some remarkable properties of McShane’s functional, defined by means of positive isotonic linear functionals. These properties are then applied to weighted generalized means. A series of consequences among additive and multiplicative type mean inequalities is given, as well as a special consideration of Holder’s inequality, in view of the new results.

Extensions of Representable Positive Linear Functionals to Unitized Quasi *-Algebras

It is known that, under certain conditions, *-representability and extensibility to the unitized *-algebra of a positive linear functional, defined on a *-algebra without unit, are equivalent. In this paper, a new condition for an analogous result is given for the case of a hermitian linear functional defined on a quasi *-algebra \({(\mathfrak{A}, \mathfrak{A}_0)}\) without unit.

A note on triangle equality for C*-seminorms and positive linear functionals on Banach *-modules

Let X be a semi-inner product module over a Banach ∗-algebra A and γ a C ∗ -seminorm or a positive linear functional on A, we define a real function Γ on X by Γ(x )=( γ(� x, x� )) 1/2 . we find necessary and sufficient conditions for the triangle equality Γ(x + y )=Γ (x )+Γ (y) of elements x and y in a semi-inner product module over unital Banach ∗-algebras, unital hermitian Banach ∗-algebras and A ∗ -algebras in terms of the numerical range. Mathematics Subject Classification: 47A30; 46L08; 47A12

A Remark on Continuity of Positive Linear Functionals on Separable Banach *-Algebras

Using a variation of the Murphy-Varopoulos Theorem, we give a new proof of the following R.J.Loy Theorem: Let A be a separable Banach*-algebra with center Z such that ZA has at most countable codimension, then every positive linear functional on A is continuous.

The Applications of Functional Variants of Jensen's Inequality

The paper is inspired by McShane's results on the functional form of Jensen's inequality for convex functions of several variables. The work is focused on applications and generalizations of this important result. At that, the generalizations of Jensen's inequality are obtained using the positive linear functionals.

The moment problem for continuous positive semidefinite linear functionals

Let $\tau$ be a locally convex topology on the countable dimensional polynomial $\reals$-algebra $\rx:=\reals[X_1,...,X_n]$. Let $K$ be a closed subset of $\reals^n$, and let $M:=M_{\{g_1, ... g_s\}}$ be a finitely generated quadratic module in $\rx$. We investigate the following question: When is the cone $\Pos(K)$ (of polynomials nonnegative on $K$) included in the closure of $M$? We give an interpretation of this inclusion with respect to representing continuous linear functionals by measures. We discuss several examples; we compute the closure of $M=\sos$ with respect to weighted norm-$p$ topologies. We show that this closure coincides with the cone $\Pos(K)$ where $K$ is a certain convex compact polyhedron.