Biseparating maps between operator algebras

A Radon-Nikodym theorem for completely n-positive linear maps on pro-C*-algebras and its applications

The order relation on the set of completely n-positive linear maps from a pro-C*-algebra A to L(H), the C*-algebra of bounded linear operators on a Hilbert space H, is characterized in terms of the representation associated with each completely n-positive linear map. Also, the pure elements in the set of all completely n-positive linear maps from A to L(H) and the extreme points in the set of unital completely n-positive linear maps from A to L(H) are characterized in terms of the representation induced by each completely n-positive linear map.

在紧元处具有乘积性质的 C*-代数上的双线性映射

Bilinear maps on C$^*$-algebras that have product property at a compact element

本文研究 C*-代数上在紧元处具有乘积性质的有界双线性映射，分别对紧 C*-代数和具有非平凡原子部分的 von Neumann 代数给出何时由该点乘积确定的判据，并应用于刻画固定点处类似同态与类似导子的映射。

Generic properties of module maps and characterizing inverse limits of C*-algebras of compact operators

We study closedness of the range, adjointability and generalized invertibility of modular operators between Hilbert modules over locally C*-algebras of coefficients. Our investigations and the recent results of M. Frank [Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules, {\it J. K-Theory} {\bf 2} (2008), 453-462] reveal a number of equivalence properties of the category of Hilbert modules over locally C*-algebras which characterize precisely the inverse limit of C*-algebras of the C*-algebra of compact operators.

On C*-Extreme Maps and *-Homomorphisms of a Commutative C*-Algebra

The generalized state space of a commutative C*-algebra, denoted S_H(C(X)), is the set of positive unital maps from C(X) to the algebra B(H) of bounded linear operators on a Hilbert space H. C*-convexity is one of several non-commutative analogs of convexity which have been discussed in this context. In this paper we show that a C*-extreme point of S_H(C(X)) satisfies a certain spectral condition on the operators in the range of the associated positive operator-valued measure. This result enables us to show that C*-extreme maps from C(X) into K^+, the algebra generated by the compact and scala…

Approximately order zero maps between C*-algebras

We investigate linear operators between C$^\ast$-algebras which approximately preserve involution and orthogonality, the latter meaning that for some $\varepsilon>0$ we have $\|ϕ(x)ϕ(y)\|\leq\varepsilon\|x\|\|y\|$ for all positive $x,y$ with $xy=0$. We establish some structural properties of such maps concerning approximate Jordan-like equations and almost commutation relations. In some situations (e.g. when the codomain is finite-dimensional), we show that $ϕ$ can be approximated by an approximate Jordan $^\ast$-homomorphism, with both errors depending only on $\|ϕ\|$ and $\varepsilon$.

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.

Linear maps behaving like derivations or anti-derivations at orthogonal elements on C*-algebras

Let A be a C*-algebra and d from A into A** be a continuous linear map. We assume that d acts like derivation or anti-derivation at orthogonal elements for several types of orthogonality conditions such as ab=0, ab*=0, ab=ba=0 and ab*=b*a=0. In each case, we characterize the structure of d. Then we apply our results for von Neumann algebras and unital simple C*-algebras.

Groupoid Characterization of Locally Convex Partial $^*$-Algebras

Given a locally convex space $(\mathcal{A},τ)$ with a Hausdorff locally convex topology $τ$ such that the following maps are continuous; $u \mapsto u^*$ for all $u \in \mathcal{A}$, $x \mapsto x\cdot y$ and $x \mapsto z\cdot x$ for every left and right multipliers of $\mathcal{A}$. In this paper we re-characterized the locally convex partial $*$-algebra $(\mathcal{A}, Γ,\cdot,*,τ)$ arising from these continuous maps in terms of convolution algebra of a Lie groupoid $Γ\rightrightarrows \mathcal{A}$. This is advantageous because the pathologies of the underlying spaces owing to their quantum mec…

Jordan maps and triple maps on algebras of unbounded operators

The paper contains results on the structure of Jordan maps and several kinds of triple maps on standard algebras of unbounded operators in Hilbert spaces. These results are unbounded counterparts to results on algebras of bounded operators obtained by Lu, Molnár and others ([3,4,6 -- 10]).

Hilbert isometries and maximal deviation preserving maps on JB-algebras

In this paper we characterize the surjective linear variation norm isometries on JB-algebras. Variation norm isometries are precisely the maps that preserve the maximal deviation, the quantum analogue of the standard deviation, which plays an important role in quantum statistics. Consequently, we characterize the Hilbert's metric isometries on cones in JB-algebras.

Projective C*-Algebras and Boundary Maps

Both boundary maps in K-theory are expressed in terms of surjections from projective C*-algebras to semiprojective C*-algebras.

A class of generalized positive linear maps on matrix algebras

We construct a class of positive linear maps on matrix algebras. We find conditions when these maps are atomic, decomposable and completely positive. We obtain a large class of atomic positive linear maps. As applications in quantum information theory, we discuss the structural physical approximation and optimality of entanglement witness associated with these maps.

Tensor products of positive maps of matrix algebras

We give conditions for when the tensor product of two positive maps between matrix algebras is a positive map. This happens when one map belongs to a symmetric mapping cone and the other to the dual cone. Necessary and sufficient conditions are given.

$*$-Lie-Jordan-type maps on $C^{*}$-algebras

Let A and A' be two Cstar-algebras with identities I_A and I_A', respectively, and P_1 and P_2 = I_A - P_1 nontrivial projections in A. In this paper we study the characterization of multiplicative star-Lie-Jordan-type maps, where the notion of these maps arise here. In particular, if M_A is a von Neumann algebra relative Cstar-algebra A without central summands of type I_1 then every bijective unital multiplicative star-Lie-Jordan-type maps are star-ring isomorphisms.

Weakly Nuclear Maps on Real $C^*$-algebras and Quasidiagonality of These Algebras

In this paper, we show that a completely positive linear map is weakly nuclear if and only if its complexification is weakly nuclear. It is shown that a real $C^*$-algebra is exact if and only if its complexification is exact and similar case is provided for the quasidiaginality.

Completely positive maps on Hilbert modules over pro-C*-algebras

We derive Paschke's GNS construction for completely positive maps on unital pro-C*-algebras from the KSGNS construction, presented by M. Joita [J. London Math. Soc. {\bf 66} (2002), 421--432], and then we deduce an analogue of Stinespring theorem for Hilbert modules over pro-C*-algebras. Also, we obtain a Radon-Nikodym type theorem for operator valued completely positive maps on Hilbert modules over pro-C*-algebras.

$*$-Jordan-type maps on $C^{*}$-algebras

Let $\mathfrak{A}$ and $\mathfrak{A}'$ be two $C^*$-algebras with identities $I_{\mathfrak{A}}$ and $I_{\mathfrak{A}'}$, respectively, and $P_1$ and $P_2 = I_{\mathfrak{A}} - P_1$ nontrivial projections in $\mathfrak{A}$. In this paper we study the characterization of multiplicative $*$-Jordan-type maps. In particular, if $\mathcal{M}$ is a factor von Neumann algebra then every bijective unital multiplicative $*$-Jordan-type maps are $*$-ring isomorphisms.

Strong Morita equivalence for completely positive linear maps on $C^*$-algebras

We will introduce the notion of strong Morita equivalence for completely positive linear maps and study its basic properties. Also, we will discuss the relation between strong Morita equivalence for bounded $C^*$-bimodule linear maps and strong Morita equivalence for completely positive linear maps. Furthermore, we will show that if two unital $C^*$-algebras are strongly Morita equivalent, then there is a $1-1$ correspondence between the two sets of all strong Morita equivalence classes of completely positive linear maps on the two unital $C^*$-algebras and we will show that the corresponding …

相对超图代数与无穷区间映射

Relative ultragraph algebras and infinite interval maps

本文提出相对超图代数，把分支系统表示的单射性判据推广到相对情形，并揭示其与无穷马尔可夫映射动力学的内在联系，为相关表示提供自然框架。

The analogue of Choi matrices for a class of linear maps on Von Neumann algebras

The definition of Choi matrices for linear maps on the n x n matrices is extended to factors, and the basic theorems for Choi matrices are proved in this general context.

C*-代数上多重线性映射的Jordan范数与Grothendieck不等式

Jordan norms for multilinear maps on C*-algebras and Grothendieck's inequalities

该研究提出C*-代数上多重线性映射完全有界范数的推广形式Jordan范数，通过映射分解定义，并证明C*-代数对上的有界双线性型均为Jordan有界，且其Jordan范数不超过原范数的两倍。

交错*-代数上的*-Lie型映射

*-Lie-Type maps on alternative *-algebras

该文研究含非平凡对称幂等元的交错*-代数上乘法*-Lie型映射的刻画，并将结果应用于交错W*-代数，丰富了非结合*-代数上映射理论的研究。

Elementary Maps on Triangular Algebras

In this note we prove that elementary maps on triangular algebras are automically additive.

Mean dimension and AH-algebras with diagonal maps

Mean dimension for AH-algebras is introduced. It is shown that if a simple unital AH-algebra with diagonal maps has mean dimension zero, then it has strict comparison on positive elements. In particular, the strict order on projections is determined by traces. Moreover, a lower bound of the mean dimension is given in term of comparison radius. Using classification results, if a simple unital AH-algebra with diagonal maps has mean dimension zero, it must be an AH-algebra without dimension growth. Two classes of AH-algebras are shown to have mean dimension zero: the class of AH-algebras with at …

Real operator algebras and real positive maps

We present some foundations for a theory of real operator algebras and real Jordan operator algebras, and the various morphisms between these. A common theme is the ingredient of real positivity from papers of the first author with Read, Neal, Wang, and other coauthors, which we import to the real scalar setting here.

Contractive projections and real positive maps on operator algebras

We study contractive projections, isometries, and real positive maps on algebras of operators on a Hilbert space. For example we find generalizations and variants of certain classical results on contractive projections on C*-algebras and JB-algebras due to Choi, Effros, Størmer, Friedman and Russo, and others. In fact most of our arguments generalize to contractive `real positive' projections on Jordan operator algebras, that is on a norm-closed space A of operators on a Hilbert space which are closedunder the Jordan product. We also prove many new general results on real positive maps which a…

完全正映射：Pro-C*-代数与Pro-C*-代数上的Hilbert模块的诱导表示

Completely Positive Maps: Pro-C*-algebras and Hilbert modules over Pro-C*-algebras

本文研究了Pro-C*-代数之间完全正映射的结构，并通过构造建立了Hilbert模块之间ϕ-映射的结构定理，同时探讨了表示的最小性。

KMS states on C*-algebras associated to expansive maps

Using Walters' version of the Ruelle-Perron-Frobenius Theorem we show the existence and uniqueness of KMS states for a certain one-parameter group of automorphisms on a C*-algebra associated to a positively expansive map on a compact metric space.

Pointwise-generalized-inverses of linear maps between C$^*$-algebras and JB$^*$-triples

We study pointwise-generalized-inverses of linear maps between C$^*$-algebras. Let $Φ$ and $Ψ$ be linear maps between complex Banach algebras $A$ and $B$. We say that $Ψ$ is a pointwise-generalized-inverse of $Φ$ if $Φ(aba)=Φ(a)Ψ(b)Φ(a),$ for every $a,b\in A$. The pair $(Φ,Ψ)$ is Jordan-triple multiplicative if $Φ$ is a pointwise-generalized-inverse of $Ψ$ and the latter is a pointwise-generalized-inverse of $Φ$. We study the basic properties of this maps in connection with Jordan homomorphism, triple homomorphisms and strongly preservers. We also determine conditions to guarantee the automati…

Weak-2-local symmetric maps on C*-algebras

We introduce and study weak-2-local symmetric maps between C$^*$-algebras $A$ and $B$ as non necessarily linear nor continuous maps $Δ: A\to B$ such that for each $a,b\in A$ and $ϕ\in B^{*}$, there exists a symmetric linear map $T_{a,b,ϕ}: A\to B$, depending on $a$, $b$ and $ϕ$, satisfying $ϕΔ(a) = ϕT_{a,b,ϕ}(a)$ and $ϕΔ(b) = ϕT_{a,b,ϕ}(b)$. We prove that every weak-2-local symmetric map between C$^*$-algebras is a linear map. Among the consequences we show that every weak-2-local $^*$-derivation on a general C$^*$-algebra is a (linear) $^*$-derivation. We also establish a 2-local version of t…

Lifting graph $C^*$-algebra maps to Leavitt path algebra maps

Let $ξ:C^*(E)\to C^*(F)$ be a unital $*$-homomorphism between simple purely infinite Cuntz-Krieger algebras of finite graphs. We prove that there exists a unital $*$-homomorphism $ϕ:L(E)\to L(F)$ between the corresponding Leavitt path-algebras such that $ξ$ is homotopic to the map $\hatϕ:C^*(E)\to C^*(F)$ induced by completion. We show moreover that $\hatϕ$ is a homotopy equivalence in the $C^*$-algebraic sense if and only if $ϕ$ is a homotopy equivalence in the algebraic, polynomial sense. We deduce, in particular, that any isomorphism between simple purely infinite Cuntz-Krieger algebras is …

将映射分类到一致迹序列代数中

Classifying maps into uniform tracial sequence algebras

本文利用迹数据对从核C*-代数到核Z-稳定C*-代数的一致迹序列代数的*-同态进行分类。

Crossed product of a C*-algebra by a semigroup of bounded positive linear maps. Interactions

The paper presents a construction of the crossed product of a C*-algebra by a commutative semigroup of bounded positive linear maps generated by partial isometries. In particular, it generalizes Antonevich, Bakhtin, Lebedev's crossed product by an endomorphism, and is related to Exel's interactions. One of the main goals is the Isomorphism Theorem established in the case of actions by endomorphisms.

Additivity of Maps on Triangular Algebras

We prove that every multiplicative bijective map, Jordan bijective map, and Jordan triple bijective map from a triangular algebra onto any ring is automatically additive.

Rank Preserving Maps on CSL Algebras

We give a description of a weakly continuous rank preserving map on a reflexive algebra on complex Hilbert space with commutative completely distributive subspace lattice. We show that the implementation of a rank preserving map can be described by the combination of two different types of maps. We also show that a rank preserving map can be implemented by only one type if the corresponding lattice is irreducible. We present some examples of both types of rank preserving map.

C*-algebras associated with interval maps

For each piecewise monotonic map tau of [0,1], we associate a pair of C*-algebras F_tau and O_tau and calculate their K-groups. The algebra F_tau is an AI-algebra. We characterize when F_tau and O_τare simple. In those cases, F_tau has a unique trace, and O_tau is purely infinite with a unique KMS-state. In the case that tau is Markov, these algebras include the Cuntz-Krieger algebras O_A, and the associated AF-algebras F_A. Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and beta-transformations. For the case of in…

Pure Maps between Euclidean Jordan Algebras

We propose a definition of purity for positive linear maps between Euclidean Jordan Algebras (EJA) that generalizes the notion of purity for quantum systems. We show that this definition of purity is closed under composition and taking adjoints and thus that the pure maps form a dagger category (which sets it apart from other possible definitions.) In fact, from the results presented in this paper, it follows that the category of EJAs with positive contractive linear maps is a dagger-effectus, a type of structure originally defined to study von Neumann algebras in an abstract categorical setti…

Linear maps on C*-algebras which are derivations or triple derivations at a point

We prove new results on generalized derivations on C$^*$-algebras. By considering the triple product $\{a,b,c\} =2^{-1} (a b^* c + c b^* a)$, we introduce the study of linear maps which are triple derivations or triple homomorphisms at a point. We prove that a continuous linear $T$ map on a unital C$^*$-algebra is a generalized derivation whenever it is a triple derivation at the unit element. If we additionally assume $T(1)=0,$ then $T$ is a $^*$-derivation and a triple derivation. Furthermore, a continuous linear map on a unital C$^*$-algebra which is a triple derivation at the unit element …

Unifying interval maps and branching systems with applications to relative graph C*-algebras

We describe Markov interval maps via branching systems and develop the theory of relative branching systems, characterizing when the associated representations of relative graph C*-algebras are faithful. When the Markov interval maps $f$ have escape sets, we use our results to characterize injectivity of the associated relative graph algebra representations, improving on previous work by the first, third, and fourth authors.

Jordan maps on standard operator algebras

Jordan isomorphisms of rings are defined by two equations. The first one is the equation of additivity while the second one concerns multiplicativity with respect to the so-called Jordan product. In this paper we present results showing that on standard operator algebras over spaces with dimension at least 2, the bijective solutions of that second equation are automatically additive.

Local maps and the representation theory of operator algebras

Using representation theory techniques we prove that various spaces of derivations or one-sided multipliers over certain operator algebras are reflexive. A sample result: any bounded local derivation (local left multiplier) on an automorphic semicrossed product is a derivation (resp. left multiplier). In the process we obtain various results of independent interest. In particular, the finite dimensional nest representations of the tensor algebra of a topological graph separate points.

New tools for investigating positive maps in matrix algebras

We provide a novel tool which may be used to construct new examples of positive maps in matrix algebras (or, equivalently, entanglement witnesses). It turns out that this can be used to prove positivity of several well known maps (such as reduction map, generalized reduction, Robertson map, and many others). Furthermore, we use it to construct a new family of linear maps and prove that they are positive, indecomposable and (nd)optimal.

K-theory for Cuntz-Krieger algebras arising from real quadratic maps

We compute the $K$-groups for the Cuntz-Krieger algebras $\mathcal{O}_{A_{\mathcal{K}(f_μ)}}$, where $A_{\mathcal{K}(f_μ)}$ is the Markov transition matrix arising from the \textit{kneading sequence }$\mathcal{K} (f_μ)$ of the one-parameter family of real quadratic maps $f_μ$.

Circle maps and C*-algebras

We consider a construction of C*-algebras from continuous piecewise monotone maps on the circle which generalizes the crossed product construction for homeomorphisms and more generally the construction of Renault, Deaconu and Anantharaman-Delaroche for local homeomorphisms. Assuming that the map is surjective and not locally injective we give necessary and sufficient conditions for the simplicity of the C*-algebra and show that it is then a Kirchberg algebra. We provide tools for the calculation of the K-theory groups and turn them into an algorithmic method for Markov maps.

On graph algebras from interval maps

We produce and study a family of representations of relative graph algebras on Hilbert spaces that arise from the orbits of points of one dimensional dynamical systems, where the underlying Markov interval maps $f$ have escape sets. We identify when such representations are faithful in terms of the transitions to the escape subintervals.

Exposedness of elementary positive maps between matrix algebras

The positive linear maps $\ad_s$ which send matrices $x$ to $s^*xs$ play important roles in quantum information theory as well as matrix theory. It was proved by Marciniak [Linear Multilinear Alg. 61 (2013), 970--975] that the map $\ad_s$ generates an exposed ray of the convex cone of all positive linear maps. In this note, we provide two alternative proofs, using Choi matrices and Woronowicz's method, respectively.

Construction of exposed indecomposable positive linear maps between matrix algebras

We construct a large class of indecomposable positive linear maps from the $2\times 2$ matrix algebra into the $4\times 4$ matrix algebra, which generate exposed extreme rays of the convex cone of all positive maps. We show that extreme points of the dual faces for separable states arising from these maps are parametrized by the Riemann sphere, and the convex hulls of the extreme points arising from a circle parallel to the equator have the exactly same properties with the convex hull of the trigonometric moment curve studied from combinatorial topology. Any interior points of the dual faces a…

$*$-Lie-type maps on $C^*$-algebras

Let $\mathfrak{A}$ and $\mathfrak{A}'$ be two $C^*$-algebras with identities $I_{\mathfrak{A}}$ and $I_{\mathfrak{A}'}$, respectively, and $P_1$ and $P_2 = I_{\mathfrak{A}} - P_1$ nontrivial symmetric projections in $\mathfrak{A}$. In this paper we study the characterization of multiplicative $*$-Lie-type maps. In particular, if $\mathcal{M}$ is a factor von Neumann algebra then every complex scalar multiplication bijective unital multiplicative $*$-Lie-type map is $*$-isomorphism.

Choi matrices, norms and entanglement associated with positive maps on matrix algebras

We study positive maps of B(K) into B(H) for finite-dimensional Hilbert spaces K and H. Our main emphasis is on how Choi matrices and estimates of their norms with respect to mapping cones reflect various properties of the maps. Special attention will be given to entanglement properties and k-positive maps, in particular tensor products of 2-positive maps. The latter problem is directly related to the question of n-copy distillability of quantum states, for which we obtain a partial result.

可分解半群与C*-代数上的D可除动态映射

Decomposable semigroups on C*-algebras and D-divisible dynamical maps

本文研究了C*-代数上可分解半群与D可除动态映射，扩展了之前关于矩阵代数上可分解动态映射的研究，并部分推广了Lindblad关于完全正半群的工作。

Nonlinear maps preserving sums of triple products on $\ast $-algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be two unital complex $\ast $-algebras such that $\mathcal{A}$ has a nontrivial projection. In this paper, we study the structure of bijective nonlinear maps $Φ:\mathcal{A}\rightarrow \mathcal{B}$ preserving sum of triple products $α_{1} abc+α_{2} a^{*}cb^{*}+α_{3} ba^{*}c +α_{4} cab^{*}+α_{5} bca+α_{6} cb^{*}a^{*},$ where the scalars $\{α_{k}\}_{k=1}^{6}$ are complex numbers satisfying some conditions.

Traces on cores of C*-algebras associated with self-similar maps

We completely classify the extreme tracial states onthe cores of the C*-algebras associated with self-similar maps on compact metric spaces. We present a complete list of them. The extreme tracial states are the union of the discrete type tracial states given by measures supported on the finite orbits of the branch points and a continuous type tracial state given by the Hutchinson measure on the original self-similar set.

Ideals of the core of C*-algebras associated with self-similar maps

We give a complete classification of the ideals of the core of the C*-algebras associated with self-similar maps under a certain condition. Any ideal is completely determined by the intersection with the coefficient algebra C(K) of the self-similar set K. The corresponding closed subset of K is described by the singularity structure of the self-similar map. In particular the core is simple if and only if the self-similar map has no branch point. A matrix representation of the core is essentially used to prove the classification.

Quantum Programs as Kleisli Maps

Furber and Jacobs have shown in their study of quantum computation that the category of commutative C*-algebras and PU-maps (positive linear maps which preserve the unit) is isomorphic to the Kleisli category of a comonad on the category of commutative C*-algebras with MIU-maps (linear maps which preserve multiplication, involution and unit). [Furber and Jacobs, 2013] In this paper, we prove a non-commutative variant of this result: the category of C*-algebras and PU-maps is isomorphic to the Kleisli category of a comonad on the subcategory of MIU-maps. A variation on this result has been used…