Juha-Pekka Pellonpää, Kari Ylinen
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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.
In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert $A$-modules over locally $C^{*}$-algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon--Nikodym theorem for this type of completely positive $n\times n$ matrices.
We introduce S-modules, generalizing the notion of Krein $C^*$-modules, where a fixed unitary replaces the symmetry of Krein $C^*$-modules. The representation theory on S-modules is explored and for a given $*$-automorphism $α$ on a $C^*$-algebra the KSGNS construction for $α$-completely positive maps is proved. An extention of this theorem for $τ$-maps is also achieved, when $τ$ is an $α$-completely positive map, along with a decomposition theorem for $\mathfrak K$-families.
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 $A\times_{α^{}}G$.
We study positive kernels on $X\times X$, where $X$ is a set equipped with an action of a group, and taking values in the set of $\mathcal A$-sesquilinear forms on a (not necessarily Hilbert) module over a $C^*$-algebra $\mathcal A$. These maps are assumed to be covariant with respect to the group action on $X$ and a representation of the group in the set of invertible ($\mathcal A$-linear) module maps. We find necessary and sufficient conditions for extremality of such kernels in certain convex subsets of positive covariant kernels. Our focus is mainly on a particular example of these kernels…
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.
Morphisms of Cuntz-Pimsner algebras from completely positive maps
本文引入由完全正映射给出左作用的正对应半范畴，并利用KSGNS构造建立到C*-对应范畴的半函子，从而研究Cuntz-Pimsner代数间的态射及其典型例子。
We prove a local convex version of Arveson's extension theorem and of Wittstock's extension theorem. Also we prove a Stinespring type theorem for unbounded local completely contractive maps.
Let X and Y be Banach spaces with dim X greater than 3. Let A and B be standard operator algebras on X and Y. We characterize the form of maps from A onto B such that completely preserve involution.
We introduce a preorder relation in the collection of all operator valued completely positive maps on a full Hilbert C*-module and characterize this relation in terms of the Stinespring construction associated to each completely positive map.
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.
052<p type="texpara" tag="Body Text" et="abstract" >A completely $n$ -positive linear map from a locally $C^{\ast}$-algebra $A$ to another locally $C^{\ast}$-algebra $B $is an $n\times n$ matrix whose elements are continuous linear maps from $A$ to $B$ and which verifies the condition of completely positivity. In this paper we prove a Radon-Nikodym type theorem for strict completely $n$-positive linear maps which describes the order relation on the set of all strict completely $n$ -positive linear maps from a locally $C^{\ast }$-algebra $A$ to a $C^{\ast}$-algebra $B$, in terms of a self-dual …
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.
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…
Let $\mathcal{A}$ and $\mathcal{B}$ be standard operator algebras on Banach spaces $\mathcal{X}$ and $\mathcal{Y}$, respectively. In this paper, we show that every bijection completely preserving quadratic operators from $\mathcal{A}$ onto $\mathcal{B}$ is either an isomorphism or (in the complex case) a conjugate isomorphism.
In this article, we introduce local completely positive $k$-linear maps between locally $C^{\ast}$-algebras and obtain Stinespring type representation by adopting the notion of "invariance" defined by J. Heo for $k$-linear maps between $C^{\ast}$-algebras. Also, we supply the minimality condition to make certain that minimal representation is unique up to unitary equivalence. As a consequence, we prove Radon-Nikodým theorem for unbounded operator-valued local completely positive invariant $k$-linear maps. The obtained Radon-Nikodým derivative is a positive contraction on some Hilbert space wit…
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.
We provide a description of certain invariance properties of completely bounded bimodule maps in terms of their symbols. If $\mathbb{G}$ is a locally compact quantum group, we characterise the completely bounded $L^{\infty}(\mathbb{G})'$-bimodule maps that send $C_0(\hat{\mathbb{G}})$ into $L^{\infty}(\hat{\mathbb{G}})$ in terms of the properties of the corresponding elements of the normal Haagerup tensor product $L^{\infty}(\mathbb{G}) \otimes_{σ{\rm h}} L^{\infty}(\mathbb{G})$. As a consequence, we obtain an intrinsic characterisation of the normal completely bounded $L^{\infty}(\mathbb{G})'…
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…
A map between operator spaces is called completely coarse if the sequence of its amplifications is equi-coarse. We prove that all completely coarse maps must be $\mathbb R$-linear. On the opposite direction of this result, we introduce a notion of embeddability between operator spaces and show that this notion is strictly weaker than complete $\mathbb R$-isomorphic embeddability (in particular, weaker than complete $\mathbb C$-isomorphic embeddability). Although weaker, this notion is strong enough for some applications. For instance, we show that if an infinite dimensional operator space $X$ …
A linear map $Φ:\mathbb{M}_n \to \mathbb{M}_k$ is called completely copositive if the resulting matrix $[Φ(A_{j,i})]_{i,j=1}^m$ is positive semidefinite for any integer $m$ and positive semidefinite matrix $[A_{i,j}]_{i,j=1}^m$. In this paper, we present some applications of the completely copositive maps $Φ(X)=(\mathrm{tr} X)I+X$ and $Ψ(X)= (\mathrm{tr} X)I-X$. Some new extensions about traces inequalities of positive semidefinite $3\times 3$ block matrices are included.
The paper gives an operator algebras model for the conditional monotone independence, introduced by T. Hasebe. The construction is used to prove an embedding result for the N. Muraki's monotone product of C*-algebras. Also, the formulas from the definition of conditional monotone independence are used to define the monotone product of maps which is shown to preserve complete positivity, a similar to the results from the case of free products.
Relative Kubo-Ando Means of Completely Positive Maps
本文利用Arveson的Radon-Nikodym定理，为受同一环境映射支配的完全正映射定义了相对Kubo-Ando均值，并证明了其序理论性质、几何均值的块正定性刻画以及与现有方法的等价性。
Extremal marginals of an unbounded local completely positive and local completely contractive map
研究无界局部完全正且局部完全压缩映射在极大张量积上的极值点问题，刻画了相关凸子集的极值结构。
Minimal Wittstock Envelopes of Hermitian Completely Bounded Maps
本文研究厄米完全有界映射的Wittstock包络，证明每个包络下有极小包络但不唯一，给出最小优超存在判据，并定义包络相对透视以推广Okayasu结果。
We develop a completely bounded counterpart to the non-commutative Choquet boundary of an operator space. We show how the class of completely bounded linear maps is too large to accommodate our purposes. To overcome this obstacle, we isolate the subset of completely bounded linear maps on an operator space admitting a dilation of the same norm which is multiplicative on the generated $C^*$-algebra. We view such maps as analogues of the familiar unital completely contractive maps, and we exhibit many of their structural properties. Of particular interest to us are those maps which are extremal …
We study the closure of the unitary orbit of a given point in the non-commutative Choquet boundary of a unital operator space with respect to the topology of pointwise norm convergence. This may be described more extensively as the $\ast$-representations of the $\mathrm{C}^{\ast}$-envelope that are approximately unitarily equivalent to one that possesses the unique extension property. Although these $\ast$-representations do not necessarily have the unique extension property themselves, we show that their unital completely positive extensions display significant restrictions. When the underlyi…
Let (M,N) be a pair of von Neumann algebras, or of dual operator spaces with at least one of them having property S_σ, and let T be an arbitrary completely bounded mapping on M. We present an explicit construction of an amplification of T to a completely bounded mapping on the w^*-spatial tensor product of M and N. Our approach makes essential use of the description of the predual of the latter given by Effros and Ruan in terms of the projective operator space tensor product and is further based on the concept of slice maps as introduced by Tomiyama. We discuss several properties of an amplifi…
In 2006, Arveson resolved a long-standing problem by showing that for any element $x$ of a separable self-adjoint unital subspace $S\subseteq B(H)$, $\|x\|=\sup\|π(x)\|$, where $π$ runs over the boundary representations for $S$. Here we show that "sup" can be replaced by "max". This implies that the Choquet boundary for a separable operator system is a boundary in the classical sense; a similar result is obtained in terms of pure matrix states when $S$ is not assumed to be separable. For matrix convex sets associated to operator systems in matrix algebras, we apply the above results to improve…
We consider contractive homomorphisms of a planar algebra ${\mathcal A}(Ω)$ over a finitely connected bounded domain $Ω\subseteq \C$ and ask if they are necessarily completely contractive. We show that a homomorphism $ρ:{\mathcal A}(Ω) \to {\mathcal B}(\mathcal H)$ for which $\dim({\mathcal A}(Ω)/\ker ρ) = 2$ is the direct integral of homomorphisms $ρ_T$ induced by operators on two dimensional Hilbert spaces via a suitable functional calculus $ρ_T: f \mapsto f(T), f\in {\mathcal A}(Ω)$. It is well-known that contractive homomorphisms $ρ_T$, induced by a linear transformation $T:\C^2 \to \C^2$ …
On numerical diameters and linear maps
本文研究希尔伯特空间上有界算子的数值域直径所诱导的数值直径半范数，证明完全有界数值直径是与完全有界范数可比较但不相等的范数，并指出其在单位完全正映射及其截面中具有特殊意义。
Let $ϕ:M_n\to B(H)$ be an injective, completely positive contraction with $\Vϕ^{-1}:ϕ(M_n)\to M_n\V_{cb}\leq1+δ(ε).$ We show that if either (i) $ϕ(M_n)$ is faithful modulo the compact operators or (ii) $ϕ(M_n)$ approximately contains a rank 1 projection, then there is a complete order embedding $ψ:M_n\to B(H)$ with $\Vϕ-ψ\V_{cb}<ε.$ We also give examples showing that such a perturbation does not exist in general. As an application, we show that every $C^*$-algebra $A$ with $\mathcal{OL}_\infty(A)=1$ and a finite separating family of primitive ideals is a strong NF algebra, providing a partial …
$C^*$-extreme contractive completely positive maps
本文推广$C^*$-代数广义状态空间的量子凸性，引入$P$-$C^*$-凸集并刻画其极值点，完全表征有限维情形下压缩完全正映射的$C^*$-极值点，并讨论与线性极值点及Krein-Milman型定理的联系。
We initiate the study of the completely bounded multipliers of the Haagerup tensor product $A(G)\otimes_{\rm h} A(G)$ of two copies of the Fourier algebra $A(G)$ of a locally compact group $G$. If $E$ is a closed subset of $G$ we let $E^{\sharp} = \{(s,t) : st\in E\}$ and show that if $E^{\sharp}$ is a set of spectral synthesis for $A(G)\otimes_{\rm h} A(G)$ then $E$ is a set of local spectral synthesis for $A(G)$. Conversely, we prove that if $E$ is a set of spectral synthesis for $A(G)$ and $G$ is a Moore group then $E^{\sharp}$ is a set of spectral synthesis for $A(G)\otimes_{\rm h} A(G)$. …
Let G be a locally compact group, M(G) denote its measure algebra and L^1(G) denote its group algebra. Also, let pi:G->U(H) be a strongly continuous unitary representation, and let CB^{sigma}(B(H)) be the space of normal completely bounded maps on B(H). We study the range of the map Gamma_pi:M(G)->CB^sigma(B(H)), Gamma_pi(mu)= int_G pi(s)\otimes pi(s)^*dmu(s) where we identify CB^sigma(B(H)) with the extended Haagerup tensor product B(H)\otimes^{eh}B(H)$. We use the fact that the C*-algebra generated by integrating pi to L^1(G) is unital exactly when pi is norm continuous to show that Gamma_pi…
It is well-known that if T is a D_m-D_n bimodule map on the m by n complex matrices, then T is a Schur multiplier and $\|T\|_{cb}=\|T\|$. If n=2 and T is merely assumed to be a right D_2-module map, then we show that $\|T\|_{cb}=\|T\|$. However, this property fails if m>1 and n>2. For m>1 and n=3,4 or $n\geq m^2$, we give examples of maps T attaining the supremum C(m,n)=\sup \|T\|_{cb} taken over the contractive, right D_n-module maps on M_{m,n}, we show that C(m,m^2)=\sqrt{m} and succeed in finding sharp results for C(m,n) in certain other cases. As a consequence, if H is an infinite-dimensio…
Let $(M,Γ)$ be a Hopf--von Neumann algebra, so that $M_\ast$ is a completely contractive Banach algebra. We investigate whether the product of two elements of $M$ that are both weakly almost periodic functionals on $M_\ast$ is again weakly almost periodic. For that purpose, we establish the following factorization result: If $M$ and $N$ are injective von Neumann algebras, and if $x, y \in M \bar{\otimes} N$ correspond to weakly compact operators from $M_\ast$ to $N$ factoring through reflexive operator spaces $X$ and $Y$, respectively, then the operator corresponding to $xy$ factors through th…
We study the dual relationship between quantum group convolution maps $L^1(\mathbb{G})\rightarrow L^{\infty}(\mathbb{G})$ and completely bounded multipliers of $\widehat{\mathbb{G}}$. For a large class of locally compact quantum groups $\mathbb{G}$ we completely isomorphically identify the mapping ideal of row Hilbert space factorizable convolution maps with $M_{cb}(L^1(\widehat{\mathbb{G}}))$, yielding a quantum Gilbert representation for completely bounded multipliers. We also identify the mapping ideals of completely integral and completely nuclear convolution maps, the latter case coincidi…
Rescaling of unconditional Schauder frames in Hilbert spaces and completely bounded maps
本文证明Hilbert空间中每个元素可表示为无条件收敛级数时，存在非零标量使两个序列均为框架，并等价于秩一有界线性映射的完全有界性。
We prove a factorization of completely bounded maps from a $C^*$-algebra $A$ (or an exact operator space $E\subset A$) to $\ell_2$ equipped with the operator space structure of $(C,R)_θ$ ($0<θ<1$) obtained by complex interpolation between the column and row Hilbert spaces. More precisely, if $F$ denotes $\ell_2$ equipped with the operator space structure of $(C,R)_θ$, then $u: A \to F$ is completely bounded iff there are states $f,g$ on $A$ and $C>0$ such that \[ \forall a\in A\quad \|ua\|^2\le C f(a^*a)^{1-θ}g(aa^*)^θ.\] This extends the case $θ=1/2$ treated in a recent paper with Shlyakhtenk…
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。
Russo-Dye type Theorem, Stinespring representation,and Radon Nikodym dervative for invariant block multilinear completely positive maps
本文研究了不变多重线性完全正映射的性质，证明了其 Russo-Dye 型定理、自动对称性与完全有界性，引入了不变块 CP 映射的多重线性版本，推导出极小 Stinespring 膨胀并获得了 Radon Nikodym 定理。
We study the structure of nilpotent completely positive maps in terms of Choi-Kraus coefficients. We prove several inequalities, including certain majorization type inequalities for dimensions of kernels of powers of nilpotent completely positive maps.
We introduce weighted cb maps and $Λ_μ$-cb maps on operator spaces which are generalizations of completely bounded maps and a certain class of bilinear maps on operator spaces which we call $λ_μ$-cb bilinear maps. Some basic properties of these maps and an operator space tensor product associated to $λ_μ$-cb bilinear maps have been studied.
Every homomorphism $φ: B(G) \rightarrow B(H)$ between Fourier-Stieltjes algebras on locally compact groups $G$ and $H$ is determined by a continuous mapping $α: Y \rightarrow Δ(B(G))$, where $Y$ is a set in the open coset ring of $H$ and $Δ(B(G))$ is the Gelfand spectrum of $B(G)$ (a $*$-semigroup). We exhibit a large collection of maps $α$ for which $φ=j_α: B(G) \rightarrow B(H)$ is a completely positive/completely contractive/completely bounded homomorphism and establish converse statements in several instances. For example, we fully characterize all completely positive/completely contractiv…
We introduce completely semi-$φ$-maps on Hilbert $C^*$-modules as a generalization of $φ$-maps. This class of maps provides examples of CP-extendable maps which are not CP-H-extendable, in Skeide-Sumesh's sense. Using the CP-extendability of completely semi-$φ$-maps, we give a representation theorem, similar to Stinespring's representation theorem, for this class of maps which can be considered as strengthened and generalized form of Asadi's and Bhat-Ramesh-Sumesh's analogues of Stinespring representation theorem for $φ$-maps. We also define an order relation on the set of all completely semi-…
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 …
We investigate certain classes of normal completely positive (CP) maps on the hyperfinite $II_1$ factor $\mathcal A$. Using the representation theory of a suitable irrational rotation algebra, we propose some computable invariants for such CP maps.
We explain how to find the KK-theoretic counterpart of extremal K-set defined by Larry Brown and Gert Pedersen.
Metrics on completely positive maps via noncommutative geometry
本文研究通过非交换几何中的半范数诱导完全正映射的度量方法，提出无限维C*-代数的Choi-Jamiołkowski同构的类比，并展示其在量子信息理论中的稳定性与链式性质。
It is shown that a fixed point of a completely positive map on a semi-finite von Neumann algebra must commute with the operators determining the map (the Lueders phenomenon) if the element is finite or has finite square.
We will consider the set of all completely positive linear maps from a unital $C^*$-algebra to the $C^*$-algebra of all (bounded) adjointable right Hilbert $C^*$-module maps, which are automatically bounded, on a right Hilbert $C^*$-module and we will introduce strong Morita equivalence for elements in this set. In this paper, we will give the following result: If two classes of two unital $C^*$-algebras are strongly Morita equivalent, respectively, then we can construct a bijective correspondence between two sets of all strong Morita equivalence classes of completely positive linear maps give…
In this note we prove that there exist at least two examples of three commuting, unital, completely positive maps that have no dilation on a type I factor, and no minimal dilation on any von Neumann algebra.
We consider derivations from the image of the canonical contraction $θ_A$ from the Haagerup tensor product of a C*-algebra A with itself to the space of completely bounded maps on A. We show that such derivations are necessarily inner if A is prime or if A is quasicentral with Hausdorff primitive spectrum. We also provide an example of a C*-algebra which has outer elementary derivations.
Two notions for linear maps (operational convex combinations and operational exetreme points) are introduced. The set S of ucp maps on the n times n matrix algebra is the operational convex combinations of the identity map. An operational extreme point of S is an extreme point of S but the converse does not hold, and every automorphism is an operational extreme point of S.
We establish a reformulation of the Connes embedding problem in terms of an asymptotic property of factorizable completely positive maps. We also prove that the Holevo-Werner channels W_n^- are factorizable, for all odd integers n different from 3. Furthermore, we investigate factorizability of convex combinations of W_3^+ and W_3^-, a family of channels studied by Mendl and Wolf, and discuss asymptotic properties for these channels.
Quasi-invariant lifts of completely positive maps for groupoid actions
本文研究了群作用下完全正映射的准不变提升，并构造了G作用的Busby不变量。
We define the graph product of unital completely positive maps on a universal graph product of unital C*-algebras and show that it is unital completely positive itself. To accomplish this, we introduce the notion of the non-commutative length of a word, and we obtain a Stinespring construction for concatenation. This result yields the following consequences. The graph product of positive-definite functions is positive-definite. A graph product version of von Neumann's Inequality holds. Graph independent contractions on a Hilbert space simultaneously dilate to graph independent unitaries.
Anar A. Dosiev in [Local operator spaces, unbounded operators and multinormed $C^*$-algebras, J. Funct. Anal. 255 (2008), 1724-1760], obtained a Stinespring's theorem for local completely positive maps (in short: local CP-maps) on locally $C^{\ast}$-algebras. In this article a suitable notion of minimality for this construction has been identified so as to ensure uniqueness up to unitary equivalence for the associated representation. Using this a Radon-Nikodym type theorem for local completely positive maps has been proved. Further, a Stinespring's theorem for unbounded operator valued local c…
This article proves the existence of completely positive quasimultiplicative maps from the group algebra of imprimitive reflection groups to the set of bounded operators, and uses those linear maps to define creation and annihilation operators on the full Fock space.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。