Related papers: Spectral geometric mean and trace characterization…

Let $A(x)=A\_0+x\_1A\_1+...+x\_nA\_n$ be a linear matrix, or pencil, generated by given symmetric matrices $A\_0,A\_1,...,A\_n$ of size $m$ with rational entries. The set of real vectors x such that the pencil is positive semidefinite is a…

The concept of the {\em half density matrix} is proposed. It unifies the quantum states which are described by density matrices and physical processes which are described by completely positive maps. With the help of the half-density-matrix…

We present some properties of (not necessarily linear) positive maps between $C^*$-algebras. We first extend the notion of Lieb functions to that of Lieb positive maps between $C^*$-algebras. Then we give some basic properties and…

Let $A,B\in \mathbb{B}(\mathscr{H})$ be such that $0<b_{1}I \leq A \leq a_{1}I$ and $0<b_{2}I \leq B \leq a_{2}I$ for some scalars $0<b_{i}< a_{i},\;\; i=1,2$ and $\Phi:\mathbb{B}(\mathscr{H})\rightarrow\mathbb{B}(\mathscr{K})$ be a…

Let (A,phi) be the reduced free product of infinitely many pairs (A_i,phi_i) of C*-algebras with faithful states. Assume that the A_i are not too small, in a specific sense. It is shown that if phi is a trace then K_0(A) is determined…

Denote $\Sigma_n$ and $Q_n$ the set of all $n \times n$ symmetric and skew-symmetric matrices over a field $\mathbb{F}$, respectively, where $char(\mathbb{F})\neq 2$ and $\lvert \mathbb{F} \rvert \geq n^2+1$. A characterization of…

A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set…

Matrices are the most common representations of graphs. They are also used for the representation of algebras and cluster algebras. This paper shows some properties of matrices in order to facilitate the understanding and locating…

Let $\mathscr{A}$ be a unital $C^*$-algebra and let $\Phi: \mathscr{A} \to {\mathbb B}({\mathscr H})$ be a unital $n$-positive linear map between $C^*$-algebras for some $n \geq 3$. We show that $$\|\Phi(AB)-\Phi(A)\Phi(B)\| \leq…

Assuming a unitarily invariant norm $|||\cdot|||$ is given on a two-sided ideal of bounded linear operators acting on a separable Hilbert space, it induces some unitarily invariant norms $|||\cdot|||$ on matrix algebras $\mathcal{M}_n$ for…

We study trace functions on the form $ t\to\tr f(A+tB) $ where $ f $ is a real function defined on the positive half-line, and $ A $ and $ B $ are matrices such that $ A $ is positive definite and $ B $ is positive semi-definite. If $ f $…

For $q\in\mathbb{R}$, the $Q$-matrix $Q=Q_q$ of a connected simple graph $G=(V,E)$ is $Q_q=(q^{\partial(x,y)})_{x,y\in V}$, where $\partial$ denotes the path-length distance. Describing the set $\pi(G)$ consisting of those $q\in \mathbb{R}$…

Certain reduced free products of C*-algebras, (A,phi)=(A_1,phi_1)*(A_2,\phi_2), taken with respect to faithful states, at least one of which is not a trace, are shown to be purely infinite and simple. It is assumed that one of the A_i…

A real matrix is said to be positive if its every entry is positive, and a real square matrix A is algebraically positive if there exists a real polynomial f such that f(A) is a positive matrix. A sign pattern matrix A is said to require a…

We study the differential geometric properties of the manifold of non-singular symmetric real matrices endowed with the trace metric; in case of positive definite matrices we describe the full group of isometries

There are various generalizations of the geometric mean $(a,b)\mapsto a^{1/2}b^{1/2}$ for $a,b\in \mathbb{R}^+$ to positive matrices, and we consider the standard positive geometric mean $(X,Y)\mapsto…

The completely bounded trace and spectral norms in finite dimensions are shown to be expressible by semidefinite programs. This provides an efficient method by which these norms may be both calculated and verified, and gives alternate…

Symplectic eigenvalues are conventionally defined for symmetric positive-definite matrices via Williamson's diagonal form. Many properties of standard eigenvalues, including the trace minimization theorem, are extended to the case of…

The independence number of a square matrix $A$, denoted by $\alpha(A)$, is the maximum order of its principal zero submatrices. Let $S_n^{+}$ be the set of $n\times n$ nonnegative symmetric matrices with zero trace. Denote by $J_n$ the…

Following a result of Hatori, Miura and Tagaki ([4]) we give here a spectral characterization of an isomorphism from a $C^\star$-algebra onto a Banach algebra. We then use this result to show that a $C^\star$-algebra $A$ is isomorphic to a…