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.

Lebesgue Decomposition of Non-Commutative Measures

We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

Functional continuity of topological algebras with orthonormal bases

(i) Every complex [Formula: see text]-algebra with an identity and with an orthonormal basis is functionally continuous; (ii) Every complex complete LMC algebra with an orthogonal basis is functionally continuous; (iii) Every complex sequentially complete locally convex algebra with an unconditional orthonormal basis and with an element [Formula: see text] for which [Formula: see text]th coefficient functional value tends to infinity as [Formula: see text] tends to infinity is functionally continuous. These results are proved and an example is provided for non-extendability of these results. A representation for positive linear functionals on a sequentially complete locally convex algebra with an unconditional orthonormal basis, with an identity, and with an element [Formula: see text] mentioned in (iii) is obtained. All results are obtained only for commutative algebras.

INEQUALITIES OF JENSEN’S TYPE FOR POSITIVE LINEAR FUNCTIONALS ON HERMITIAN UNITAL BANACH -ALGEBRAS

We establish inequalities of Jensen’s and Slater’s type in the general setting of a Hermitian unital Banach $\ast$-algebra, analytic convex functions and positive normalised linear functionals.