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Every positive multilinear map between C∗-algebras is separately weak∗-continuous. We show that the joint weak∗-continuity is equivalent to the joint weak∗-continuity of the multiplications of the C∗-algebras under consideration. We study the behavior of general tracial positive maps on properly infinite von Neumann algebras and by applying the Aron–Berner extension of multilinear maps, we establish that under some mild conditions every tracial positive multilinear map between general C∗-algebras enjoys a decomposition Φ=φ2∘φ1, in which φ1is a tracial positive linear map with the commutative range and φ2is a tracial completely positive map with the commutative domain. As an immediate consequence, tracial positive multilinear maps are completely positive. Furthermore, we prove that if the domain of a general tracial completely positive map Φbetween C∗-algebra is a von Neumann algebra, then Φhas a similar decomposition. As an application, we investigate the generalized variance and covariance in quantum mechanics for arbitrary positive maps. Among others, an uncertainty relation inequality for commuting observables in a composite physical system is presented.

In this paper, we aim to replace in the definitions of covariance and correlation the usual trace Tr by a tracial positive map between unital C∗-algebras and to replace the functions xαand x1-αby functions fand gsatisfying some mild conditions. These allow us to define the generalized covariance, the generalized variance, the generalized correlation and the generalized Wigner–Yanase–Dyson skew information related to the tracial positive maps and functions fand g. We persent a generalization of Heisenberg’s uncertainty relation in the noncommutative framework. We extend some inequalities and properties for the generalized correlation and the generalized Wigner–Yanase–Dyson skew information. Furthermore, we extend some inequalities for the generalized skew information such as uncertainty relation and the relation between the generalized variance and the generalized skew information.

The aim of this paper is to present a comprehensive study of operator m-convex functions. Let m∈[0,1]{m\in[0,1]}, and J=[0,b]{J=[0,b]}for some b∈ℝ{b\in\mathbb{R}}or J=[0,∞){J=[0,\infty)}. A continuous function φ:J→ℝ{\varphi\colon J\to\mathbb{R}}is called operator m-convex if for any t∈[0,1]{t\in[0,1]}and any self-adjoint operators A,B∈𝔹⁢(ℋ){A,B\in\mathbb{B}({\mathscr{H}})}, whose spectra are contained in J, we have φ⁢(t⁢A+m⁢(1-t)⁢B)≤t⁢φ⁢(A)+m⁢(1-t)⁢φ⁢(B){\varphi(tA+m(1-t)B)\leq t\varphi(A)+m(1-t)\varphi(B)}. We first generalize the celebrated Jensen inequality for continuous m-convex functions and Hilbert space operators and then use suitable weight functions to give some weighted refinements. Introducing the notion of operator m-convexity, we extend the Choi–Davis–Jensen inequality for operator m-convex functions. We also present an operator version of the Jensen–Mercer inequality for m-convex functions and generalize this inequality for operator m-convex functions involving continuous fields of operators and unital fields of positive linear mappings. Employing the Jensen–Mercer operator inequality for operator m-convex functions, we construct the m-Jensen operator functional and obtain an upper bound for it.

We extend an operator Pólya–Szegö type inequality involving the operator geometric mean to any arbitrary operator mean under some mild conditions. Utilizing the Mond–Pečarić method, we present some other related operator inequalities as well. [ABSTRACT FROM AUTHOR]

Suppose that Eand Fare Hilbert C∗-modules. We present a power-norm ·nE:n∈Nbased on Eand obtain some of its fundamental properties. We introduce a new definition of the absolutely (2, 2)-summing operators from Eto F, and denote the set of such operators by Π~2(E,F)with the convention Π~2(E)=Π~2(E,E). It is known that the class of all Hilbert–Schmidt operators on a Hilbert space His the same as the space Π~2(H). We show that the class of Hilbert–Schmidt operators introduced by Frank and Larson coincides with the space Π~2(E)for a countably generated Hilbert C∗-module Eover a unital commutative C∗-algebra. These results motivate us to investigate the properties of the space Π~2(E,F).

We study the operator Q-class functions, present some Hermite- Hadamard inequalities for operator Q-class functions and give some Kantorovich and Jensen type operator inequalities involving Q-class functions

We introduce some measures of the dependence such as the strong mixing and uniform mixing coefficients in von Neumann algebras and then define the noncommutative strong and uniform mixing sequences. We establish some notable nonncommutative mixing inequalities such as Ibragimov inequality. Moreover, we extend the notion of mixingale sequence to the noncommutative content and demonstrate a noncommutative L1and weak law of large numbers for uniformly integrable L1-mixingale sequences. In addition, we investigate the noncommutative Lp-near-epoch dependence and provide some conditions under which a noncommtative Lp-near-epoch dependent sequence is a noncommutative Lp-mixingale. Finally, we introduce the concept of noncommutative Lp-approximability and show that in the setting of quantum (noncommutative) probability spaces, an Lr-bounded and L0-approximable sequence is Lp-approximable for 1≤pand 2p

Abstract: First, we take a historical glimpse at some significant refinements and extensions of the Kantorovich inequality. Second, we present some operator Kantorovich inequalities involving unital positive linear mappings and the operator geometric mean in the framework of semi-inner product -modules and give some new and classical results in a unified approach. [Copyright &y& Elsevier]

. We show first that for each C*-algebra A, contractibility of A implies contractibility of Mn (A). We next prove that an incidence algebra A of upper triangular matrices, defined by a partially ordered set Ω on {1, 2,...,n} satisfying (p, q) ∈ Ω → p ≤ q, is a contractible Banach algebra iff there is no discordant couple of D-transitive triples of elements of Ω.