On the structure of positive maps between matrix algebras

Banach双模值正映射：不等式与诱导表示

Banach bimodule-valued positive maps: Inequalities and induced representations

本文研究了Banach双模值正映射的诱导表示，并推导了相关不等式。

Biseparating maps between operator algebras

We prove that a biseparating map between spaces B(E), and some other Banach algebras, is automatically continuous and an algebra isomorphism.

Bures Distance For Completely Positive Maps

D. Bures had defined a metric on the set of normal states on a von Neumann algebra using GNS representations of states. This notion has been extended to completely positive maps between $C^*$-algebras by D. Kretschmann, D. Schlingemann and R. F. Werner. We present a Hilbert $C^*$-module version of this theory. We show that we do get a metric when the completely positive maps under consideration map to a von Neumann algebra. Further, we include several examples and counter examples. We also prove a rigidity theorem, showing that representation modules of completely positive maps which are close…

A Radon-Nikodym type theorem for $α$-completely positive maps on groups

We show that an operator valued $α$-completely positive map on a group G is given by a unitary representation of G on a Krein space which satisfies some condition. Moreover, two unitary equivalent such unitary representations define the same α-completely positive map. Also we introduce a pre-order relation on the collection of α-completely positive maps on a group and we characterize this relation in terms of the unitary representation associated to each map.

Modules, completely positive maps, and a generalized KSGNS construction

A very general KSGNS type dilation theorem in the context of right (not necessarily Hilbert) modules over $C^*$-algebras is presented. The proof uses Kolmogorov type decompositions for positive-definite kernels with values in spaces of sesquilinear maps. More specific functional analytic applications are obtained by adding assumptions.

Covariant completely positive linear maps between locally C*-algebras

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 completely positive linear map on the crossed product $A\times_{α^{}}G$.

Asymptotic lifts of positive linear maps

We show that the notion of asymptotic lift generalizes naturally to normal positive maps $ϕ$ acting on von Neumann algebras M. We focus on cases in which the domain of the asymptotic lift can be embedded as an operator subsystem of M, and characterize when that subsystem is a Jordan subalgebra of M in terms of the asymptotic multiplicative properties of $ϕ$.

Completely positive maps of order zero

We say a completely positive contractive map between two C*-algebras has order zero, if it sends orthogonal elements to orthogonal elements. We prove a structure theorem for such maps. As a consequence, order zero maps are in one-to-one correspondence with *-homomorphisms from the cone over the domain into the target algebra. Moreover, we conclude that tensor products of order zero maps are again order zero, that the composition of an order zero map with a tracial functional is again a tracial functional, and that order zero maps respect the Cuntz relation, hence induce ordered semigroup morph…

On the fixed point spaces of some completely positive maps

In this paper we generalize the results shown by Das and Peterson. Let $M$ be a ${\rm II}_1$-factor acting on $L^2(M)$. We consider certain unital normal completely positive maps on $B(L^2(M))$ which are identity on $M$. We investigate their fixed point spaces and obtain a rigidity result. As an application, we show some results of subfactors.

The eigenvectors of semigroups of positive maps on von Neumann algebras

The eigenvectors of an ergodic semigroup of linear normal positive unital maps on a von Neumann algebra are described. Moreover, it is shown by means of examples, that mere positivity of the maps in question is not sufficient for Frobenius theory as in S. Albeverio and R. H\{o}egh-Krohn, \emph{Frobenius theory of positive maps of von Neumann algebras}, Comm. Math. Phys. \textbf{64} (1978), 83--94, to hold.

Non-linear positive maps between $C^*$-algebras

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 fundamental inequalities related to such maps. Next, we study $n$-positive maps ($n\geq 2$). We show that if for a unital $3$-positive map $Φ: \mathscr{A}\longrightarrow\mathscr{B}$ between unital $C^*$-algebras and some $ A\in \mathscr{A}$ equality $Φ(A^*A)= Φ(A)^* Φ(A)$ holds, then $Φ(XA)=Φ(X)Φ(A)$ for all $X \in \mathscr{A}$. In addition, we prove that for a certa…

Maps on positive cones in operator algebras preserving power means

In this paper we consider power means of positive Hilbert space operators both in the conventional and in the Kubo-Ando senses. We describe the corresponding isomorphisms (bijective transformations respecting those means as binary operations) on positive definite cones and on positive semidefinite cones in operator algebras. We also investigate the question when those two sorts of power means can be transformed into each other.

$C^*$-代数上的乘法谱保持映射

Multiplicatively spectrum-preserving maps on $C^{*}$-algebras

本文刻画含单位元$C^*$-代数自伴元之间的乘法谱保持满射，证明其由Jordan同构与中心对称决定，并解答Molnár的问题。

A Variant of the Kaplansky Problem for Maps on Positive Matrices

We prove that all injective maps on positive complex matrices which preserve order and shrink spectrum are implemented by unitary or antiunitary conjugations. We show by counterexamples that all assumptions are indispensable. The result easily generalizes to maps on hermitian matrices.

Indecomposable exposed positive bi-linear maps between two by two matrices

Positive bi-linear maps between matrix algebras play important roles to detect tri-partite entanglement by the duality between bi-linear maps and tri-tensor products. We exhibit indecomposable positive bi-linear maps between $2\times 2$ matrices which generate extreme rays in the cone of all positive bi-linear maps. In fact, they are exposed, and so detect entanglement of positive partial transpose whose volume is nonzero.

Positive linear maps and eigenvalue estimates for nonnegative matrices

We show how positive unital linear maps can be used to obtain some bounds for the eigenvalues of nonnegative matrices.

Some generalisations of the inequalities for positive linear maps

We obtain generalisations of some inequalities for positive unital linear maps on matrix algebra. This also provides several positive semidefinite matrices and we get some old and new inequalities involving the eigenvalues of a Hermitian matrix.

On extremal positive maps acting between type I factors

The paper is devoted to the problem of classification of extremal positive maps acting between $B(K)$ and $B(H)$ where $K$ and $H$ are Hilbert spaces. It is shown that every positive map with the property that $\rank ϕ(P)\leq 1$ for any one-dimensional projection $P$ is a rank 1 preserver. It allows to characterize all decomposable extremal maps as those which satisfy the above condition. Further, we prove that every extremal positive map which is 2-positive turns out to automatically completely positive. Finally we get the same conclusion for such extremal positive maps that $\rank ϕ(P)\leq 1…

在Wiener代数正锥上谱可加的映射

Spectrally additive maps on the positive cones of the Wiener algebra

本文研究了在Wiener代数正锥上保持谱的映射，并证明其可扩展为代数的等距实线性同构。

Convex maps on R^n and positive definite matrices

We study various convex functions on $R^n$ associated with positive definite matrices. This yiels some exotic Holder matrix inequalities.

Counterexamples to the extendibility of positive unital norm-one maps

Arveson's extension theorem guarantees that every completely positive map defined on an operator system can be extended to a completely positive map defined on the whole C*-algebra containing it. An analogous statement where complete positivity is replaced by positivity is known to be false. A natural question is whether extendibility could still hold for positive maps satisfying stronger conditions, such as being unital and norm 1. Here we provide three counterexamples showing that positive norm-one unital maps defined on an operator subsystem of a matrix algebra cannot be extended to a posit…

Positive Linear Maps and Perturbation bounds of Matrices

We show how positive unital linear maps can be used to obtain lower bounds for the maximum distance between the eigenvalues of two normal matrices. Some related bounds for the spread and condition number of Hermitian matrices are also discussed here.

外围自同构保单位完全正映射

Peripherally automorphic unital completely positive maps

本文研究有限维C*-代数上外围特征向量空间中Choi-Effros积与原积一致的UCP映射，刻画其结构、分解为持久与瞬态部分，并证明谱在单位圆上的UCP映射为*-自同构。

There are many more positive maps than completely positive maps

A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool are real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps which are not completely positive is given.

On the Russo-Dye Theorem for positive linear maps

We revisit a classical result, the Russo-Dye Theorem, stating that every positive linear map attains its norm at the identity.

On Pompeiu-Cebysev type inequalities for positive linear maps of selfadjoint operators in inner product spaces

In this work, generalizations of some inequalities for continuous $h$-synchronous ($h$-asynchronous) functions of linear bounded selfadjoint operators under positive linear maps in Hilbert spaces are proved.

Korovkin-type results on convergence of sequences of positive linear maps on function spaces

In this paper we deal with the convergence of sequences of positive linear maps to a (not assumed to be linear) isometry on spaces of continuous functions. We obtain generalizations of known Korovkin-type results and provide several illustrative examples.

KSGNS type construction for $α$-completely positive maps on Krein $C^{\ast}$-modules

In this paper, we investigate $Φ$-maps associated to a certain type of $α$-completely positive maps. We then prove a KSGNS (Kasparov--Stinespring--Gel'fand--Naimark--Segal) type theorem for $α$-completely positive maps on Krein $C^*$-modules and show that the minimal KSGNS construction is unique up to unitary equivalence. We also establish a covariant version of the KSGNS type theorem for a covariant $α$-completely positive map and study the structure of minimal covariant KSGNS constructions.

Maps preserving absolute continuity and singularity of positive operators

In this paper we consider the cone of all positive, bounded operators acting on an infinite dimensional, complex Hilbert space, and examine bijective maps that preserve absolute continuity in both directions. It turns out that these maps are exactly those that preserve singularity in both directions. Moreover, in some weak sense, such maps are always induced by bounded, invertible, linear- or conjugate linear operators of the underlying Hilbert space. Our result gives a possible generalization of a recent theorem of Molnar which characterizes maps on the positive cone that preserve the Lebesgu…

A decomposition theorem for positive maps, and the projection onto a spin factor

It is shown that each positive map between matrix algebras is the sum of a maximal decomposable map and an atomic map which is both optimal and co-optimal. The result is analyzed in detail for the positive projection onto a spin factor.

Criteria and new classes of k-positive maps

We study k-positive maps on operators. Proofs are given to different positivity criteria. Special attention is on positive maps arising in the study of quantum information science. Results of other researchers are extended and improved. New classes of positive maps are constructed. Some open questions are answered.

Isometries on Banach algebras of $C(Y)$-valued maps

We propose a unified approach to the study of isometries on algebras of vector-valued Lipschitz maps and those of continuously differentiable maps by means of the notion of natural $C(Y)$-valuezations that take values in unital commutative $C^*$-algebras. A precise proof of a theorem of Jarosz \cite{ja} is exhibited.

Improved operator Kantorovich and Wielandt inequalities for positive linear maps

In this paper, we improve and generalize the operator versions of Kantorovich and Wielandt inequalities for positive linear maps on Hilbert space. Our results are more extensive and precise than many previous results due to Fu and He [Linear Multilinear Algebra, doi: 10. 1080/03081087. 2014. 880432.] and Zhang [Banach J. Math. Anal., 9 (2015), no. 1, 166-172.].

Isometries on Banach algebras of vector-valued maps

We propose a unified approach to the study of isometries on algebras of vector-valued Lipschitz maps and those of continuously differentiable maps by means of the notion of admissible quadruples. We describe isometries on function spaces of some admissible quadruples that take values in unital commutative $C^*$-algebras. As a consequence we confirm the statement of \cite[Example 8]{jp} on Lipschitz algebras and show that isometries on such algebras indeed take the canonical form.

Positive linear maps on normal matrices

For a positive linear map F and a normal matrix N, we show that |F(N)| is bounded by some simple linear combinations in the unitary orbit of F(|N|). Several elegant sharp inequalities are derived, especially for the Schur product.

交叉正线性映射、正多项式与平方和

Cross-positive linear maps, positive polynomials and sums of squares

本文研究矩阵空间中的交叉正与完全交叉正线性映射，通过关联双齐次形式将其与实代数几何联系，证明比例定量界、给出3×3矩阵映射结果并提出生成非完全交叉正映射的算法。

On an operator Kantorovich inequality for positive linear maps

We improve the operator Kantorovich inequality as follows: Let $A$ be a positive operator on a Hilbert space with $0<m\le A \le M$. Then for every unital positive linear map $Φ$, \[Φ(A^{-1})^2\le (\frac{(M+m)^2}{4Mm})^2Φ(A)^{-2}.\] As a consequence, \[Φ(A^{-1})Φ(A)+Φ(A)Φ(A^{-1}) \le \frac{(M+m)^2}{2Mm}.\]

Numerical Range Inclusion, Dilation, and completely positive maps

A proof using the theory of completely positive maps is given to the fact that if $A \in M_2$, or $A \in M_3$ has a reducing eigenvalue, then every bounded linear operator $B$ with $W(B) \subseteq W(A)$ has a dilation of the form $I \otimes A$. This gives a unified treatment for the different cases of the result obtained by researchers using different techniques.

A Family of Indecomposable Positive Linear Maps based on Entangled Quantum States

We introduce a new family of indecomposable positive linear maps based on entangled quantum states. Central to our construction is the notion of an unextendible product basis. The construction lets us create indecomposable positive linear maps in matrix algebras of arbitrary high dimension.

Mapping cone of $k$-Entanglement Breaking Maps

In \cite{CMW19}, the authors introduced $k$-entanglement breaking linear maps to understand the entanglement breaking property of completely positive maps on taking composition. In this article, we do a systematic study of $k$-entanglement breaking maps. We prove many equivalent conditions for a $k$-positive linear map to be $k$-entanglement breaking, thereby study the mapping cone structure of $k$-entanglement breaking maps. We discuss examples of $k$-entanglement breaking maps and some of their significance. As an application of our study, we characterize completely positive maps that reduce…

Linear maps characterized by special products on standard operator algebras

Let A be a unital standard algebra on a complex Banach space X with dimX >1. We characterize the linear maps D; T : A --> B(X) satisfying aT(b) + D(a)b= 0 whenever a,b in A are such that ab = 0.

Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive

We investigate the set a) of positive, trace preserving maps acting on density matrices of size N, and a sequence of its nested subsets: the sets of maps which are b) decomposable, c) completely positive, d) extended by identity impose positive partial transpose and e) are superpositive. Working with the Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiolkowski isomorphism, …

Maps on positive definite operators preserving the quantum $χ_α^2$-divergence

We describe the structure of all bijective maps on the cone of positive definite operators acting on a finite and at least two-dimensional complex Hilbert space which preserve the quantum $χ_α^2$-divergence for some $α\in [0,1]$. We prove that any such transformation is necessarily implemented by either a unitary or an antiunitary operator. Similar results concerning maps on the cone of positive semidefinite operators as well as on the set of all density operators are also derived.