‎in this paper, we study some kinds of majorizations on $textbf{m}_{n}$ and their linear or strong linear preservers. also, we find the structure of linear or strong linear preservers which are multiplicative, i.e.  linear or strong linear preservers like $phi $ with the property $phi (ab)=phi (a)phi (b)$ for every $a,bin textbf{m}_{n}$.

We extend the notions of irreducibility and periodicity of a stochastic matrix to a unital positive linear map © on a finite-dimensional C∗algebra A and discuss the non-commutative version of the Perron-Frobenius theorem. For a completely positive linear map © with ©(a) = ∑ l Ll ∗aLl, we give conditions on the Ll’s equivalent to irreducibility or periodicity of ©. As an example, positive linear...

We prove a covariant version of the KSGNS (Kasparov, Stinespring, Gel’fand,Naimark,Segal) construction for completely positive linear maps between locally C-algebras. As an application of this construction, we show that a covariant completely positive linear map ρ from a locally C-algebra A to another locally C -algebra B with respect to a locally C-dynamical system (G,A,α) extends to a complet...

For any C-algebra A let A denote the set of all positive elements in A. A state on a unital C-algebra A is a linear functional ω : A → C such that ω(a) ≥ 0 for every a ∈ A and ω(I) = 1 where I is the unit of A. By S(A) we will denote the set of all states on A. For any Hilbert space H we denote by B(H) the set of all bounded linear operators on H . A linear map φ : A → B between C-algebras is c...

Let Ln be the n-dimensional second order cone. A linear map from R m to Rn is called positive if the image of Lm under this map is contained in Ln. For any pair (n, m) of dimensions, the set of positive maps forms a convex cone. We construct a linear matrix inequality of size (n−1)(m−1) that describes this cone.

For a linear map from Mm to Mn, besides the usual positivity, there are two stronger notions, complete positivity and super-positivity. Given a positive linear map φ we study a decomposition φ = φ − φ with completely positive linear maps φ (j = 1, 2). Here φ + φ is of simple form with norm small as possible. The same problem is discussed with superpositivity in place of complete positivity.

Interval observers have been the subject of an increased attention over the recent years. The vast majority of the recent works only focus on the design of such observers, but not their optimization. We partially fill this gap by considering the design of interval observers that minimize the L∞-gain of the linear operator mapping the persistent inputs to the observed outputs. While determining ...

A positive map between Euclidean Jordan algebras is a (symmetric cone) order preserving linear map. We show that the norm of such a map is attained at the unit element, thus obtaining an analog of the operator/matrix theoretic Russo-Dye theorem. A doubly stochastic map between Euclidean Jordan algebras is a positive, unital, and trace preserving map. We relate such maps to Jordan algebra automo...

We show that every entanglement with positive partial transpose may be constructed from an indecomposable positive linear map between matrix algebra.