$C^*$-extreme points of positive operator valued measures and unital completely positive maps
Tathagata Banerjee, B V Rajarama Bhat, Manish Kumar
备注：
36 pages; Some comments on a result in Holevo's book 'Statistical Structure of Quantum Theory' included after Corollary 2.10; A summary of our main results provided in the last Section; Some Remarks (2.3, 2.7, 4.6) added for clarification purposes; Several typos corrected; To appear in Communications in Mathematical Physics
We study the quantum ($C^*$) convexity structure of normalized positive operator valued measures (POVMs) on measurable spaces. In particular, it is seen that unlike extreme points under classical convexity, $C^*$-extreme points of normalized POVMs on countable spaces (in particular for finite sets) are always spectral measures (normalized projection valued measures). More generally it is shown that atomic $C^*$-extreme points are spectral. A Krein-Milman type theorem for POVMs has also been proved. As an application it is shown that a map on any commutative unital $C^*$-algebra with countable spectrum (in particular ${\mathbb C}^n$) is $C^*$-extreme in the set of unital completely positive maps if and only if it is a unital $*$-homomorphism.
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.
Using ideas due to Jean-Luc Sauvageot, we prove the existence of a continuous unital dilation of a CP$_0$-semigroup on a separable W$^*$-algebra. This paper presents the material in the author's Ph. D. thesis (arXiv.org:1304.0134.pdf) with some generalizations and an improved exposition.
Dilations of completely positive semigroups to endomorphism semigroups have been studied by numerous authors. Most existing dilation theorems involve a non-unital embedding, corresponding to the embedding of $B(H)$ as a corner of $B(K)$ for Hilbert spaces $H \subset K$. A 1986 paper of Jean-Luc Sauvageot shows how to achieve a unital dilation, but does not specify how to do so while also preserving continuity properties of the original semigroup. This thesis further develops Sauvageot's dilation theory in order to establish the existence of continuous unital dilations, and to explore connectio…
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.
In an attempt to propose more general conditions for decoherence to occur, we study spectral and ergodic properties of unital, completely positive maps on not necessarily unital $C^*$-algebras, with a particular focus on gapped maps for which the transient portion of the arising dynamical system can be separated from the persistent one. After some general results, we first devote our attention to the abelian case by investigating the unital $*$-endomorphisms of, in general non-unital, $C^*$-algebras, and their spectral structure. The finite dimensional case is also investigated in detail, and …
In this paper, we answer a question of Blecher-Muhly-Paulsen pertaining to identifying topological invariants for completely bounded Morita equivalences of holomorphic cross-section algebras. Given a certain natural subcontext of a strong Morita context of $n$-homogeneous $C^*$-algebras whose spectrum $T$ is an annulus, Blecher-Muhly-Paulsen are able to estimate the norm of a lifting of the identity of a holomorphic subalgebra by a conformal invariant of the annulus and a property of the associated matrix bundle. We give a generalization of the above example in which $T$ is a bordered Riemann …
The study of open quantum systems relies on the notion of unital completely positive semigroups on $C^*$-algebras representing physical systems. The natural generalisation would be to consider the unital completely positive semigroups on operator systems. We show that any continuous unital completely positive semigroup on matricial system can be extended to a semigroup on a finite-dimensional $C^*$-algebra, which is an injective envelope of the matricial system. In case the semigroup is invertible, this extension is unique.
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.
Consider a unital C*-algebra A, a von Neumann algebra M, a unital sub-C*-algebra C of A and a unital *-homomorphism $π$ from C to M. Let u: A --> M be a decomposable map (i.e. a linear combination of completely positive maps) which is a C-bimodule map with respect to $π$. We show that u is a linear combination of C-bimodule completely positive maps if and only if there exists a projection e in the commutant of $π(C)$ such that u is valued in eMe and $eπ(.)e$ has a completely positive extension A --> eMe.
We consider the convex set of ( unital ) positive ( completely ) maps from a $C^*$ algebra $\cla$ to a von-Neumann sub-algebra $\clm$ of $\clb(\clh)$, the algebra of bounded linear operators on a Hilbert space $\clh$ and study its extreme points via its canonical lifting to the convex set of ( unital ) positive ( complete ) normal maps from $\hat{\cla}$ to $\clm$, where $\hat{\cla}$ is the universal enveloping von-Neumann algebra over $\cla$. If $\cla=\clm$ and a ( complete ) positive operator $τ$ is a unique sum of a normal and a singular ( complete ) positive maps. Furthermore, a unital comp…
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 …
It is well known that unital contractive representations of the disk algebra are completely contractive. Let A denote the subalgebra of the disk algebra consisting of those functions f whose first derivative vanishes at 0. We prove that there are unital contractive representations of A which are not completely contractive, and furthermore provide a Kaiser and Varopoulos inspired example for A and present a characterization of those contractive representations of A which are completely contractive. In the positive direction, for the algebra of rational functions with poles off the distinguished…
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…
We demonstrate new abstract characterizations for unital and non-unital operator spaces. We characterize unital operator spaces in terms of the cone of accretive operators (operators whose real part is positive). Defining the gauge of an operator $T \in B(H)$ to be $\|Re(T)_+\|$, we demonstrate an abstract characterization of operator spaces up to complete gauge-isometry. Both of these characterizations preserve the structure of the self-adjoint, positive, and accretive operators, as well as the operator norm. We show that an operator space with a given matrix ordering of positive or accretive…
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…
In this paper, we study the multiplicative behaviour of quantum channels, mathematically described by trace preserving, completely positive maps on matrix algebras. It turns out that the multiplicative domain of a unital quantum channel has a close connection to its spectral properties. A structure theorem, which reveals the automorphic property of an arbitrary unital quantum channel on a subalgebra, is presented. Various classes of quantum channels (irreducible, primitive etc.) are then analysed in terms of this stabilising subalgebra. The notion of the multiplicative index of a unital quantu…
We prove that an operator system $\mathcal S$ is nuclear in the category of operator systems if and only if there exist nets of unital completely positive maps $ϕ_λ: \cl S \to M_{n_λ}$ and $ψ_λ: M_{n_λ} \to \cl S$ such that $ψ_λ\circ ϕ_λ$ converges to ${\rm id}_{\cl S}$ in the point-norm topology. Our proof is independent of the Choi-Effros-Kirchberg characterization of nuclear $C^*$-algebras and yields this characterization as a corollary. We give an example of a nuclear operator system that is not completely order isomorphic to a unital $C^*$-algebra.
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.
For a state $ω$ on a C$^*$-algebra $A$ we characterize all states $ρ$ in the weak* closure of the set of all states of the form $ω\circφ$, where $φ$ is a map on $A$ of the form $φ(x)=\sum_{i=1}^na_i^*xa_i,$ $\sum_{i=1}^na_i^*a_i=1$ ($a_i\in A$, $n=1,2,...$). These are precisely the states $ρ$ that satisfy $\|ρ|J\|\leq\|ω|J\|$ for each ideal $J$ of $A$. The corresponding question for normal states on a von Neumann algebra $R$ (with the weak* closure replaced by the norm closure) is also considered. All normal states of the form $ω\circψ$, where $ψ$ is a quantum channel on $R$ (that is, a map of…
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…
A unital $C^*$-algebra is called $N$-subhomogeneous if its irreducible representations are finite dimensional with dimension at most $N$. We extend this notion to operator systems, replacing irreducible representations by boundary representations. This is done by considering the matrix state space associated with an operator system and identifying the boundary representations as absolute matrix extreme points. We show that two $N$-subhomogeneous operator systems are completely order equivalent if and only if they are $N$-order equivalent. Moreover, we show that a unital $N$-positive map into a…
We construct projective limit of projective sequence in the following categories: Archimedean order unit spaces with unital positive maps and operator systems with unital completely positive maps. We prove that inductive limit and projective limit in these categories are in duality, provided that the dual objects remain in the same categories and the maps are order embeddings. We generalize a result of Choi and Effros on matrix-ordered duals of finite-dimensional operator systems to separable case.
Frames on Hilbert C*-modules have been defined for unital C*-algebras by Frank and Larson and operator valued frames on a Hilbert space have been studied in arXiv.0707.3272v1.[math.FA]. Goal of the present paper is to introduce operator valued frames on a Hilbert C*-module for a sigma-unital C*-algebra. Theorem 1.4 reformulates the definition given by Frank and Larson in terms of a series of rank-one operators converging in the strict topology. Theorem 2.2. shows that the frame transform and the frame projection of an operator valued frame are limits in the strict topology of a series of eleme…
We provide a characterization for operator valued completely bounded linear maps on Hilbert $C^*$-modules in terms of $φ$-maps. Also, we show that for every operator valued completely positive map $φ$ on a $C^*$-algebra $\mathcal{A}$, there is a unique (up to multiplication by a unitary operator) non-degenerate $φ$-map on each Hilbert $\mathcal{A}$-module.
Given a separable unital C*algebra $C$, let $E_n$ denote the Hilbert module equal to the completion of the Schwartz space of rapidly decreasing smooth functions from $R^n$ to $C$ equipped with the $C$-valued inner product given by integration. Let $B$ denote the space of all smooth functions with bounded derivatives from $R^n$ to $C$. For each $a$ in $B$, let $O(a)$ denote the pseudodifferential operator of symbol $a$. $O$ maps $B$ to $H$, the set of all adjointable operators on $E_n$ which have smooth orbit under the canonical action of the Heisenberg group. We construct a left inverse for $O…
A well-known theorem of Paulsen says that if $\mathcal{A}$ is a unital operator algebra and $ϕ:\mathcal{A}\to B(\mathcal{H})$ is a unital completely bounded homomorphism, then $ϕ$ is similar to a completely contractive map $ϕ'$. Motivated by classification problems for Hilbert space contractions, we are interested in making the inverse $ϕ'^{-1}$ completely contractive as well whenever the map $ϕ$ has a completely bounded inverse. We show that there exist invertible operators $X$ and $Y$ such that the map $$ XaX^{-1}\mapsto Yϕ(a)Y^{-1} $$ is completely contractive and is "almost" isometric on a…
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…
Let $C^*(\cls)$ be the $C^*$ algebra generated by an operator system $\cls$ i.e. a unital $*$-closed subspace of a unital $C^*$ algebra $\cla$. We prove that any complete order isomorphism $\cli:\cls \raro \cls'$ between two such operator systems of matrix algebras has a unique extension to a $C^*$-isomorphism $\cli:C^*(\cls) \raro C^*(\cls')$. However, the same statement with more general operator systems of infinite dimensional $C^*$-algebra is false. As an application of this result, we characterise upto cocycle conjugacy the extreme points of unital completely positive maps on matrix algeb…
Given a separable unital C*-algebra A, let E denote the Banach-space completion of the A-valued Schwartz space on Rn with norm induced by the A-valued inner product $<f,g>=\int f(x)^*g(x) dx$. The assignment of the pseudodifferential operator B=b(x,D) with A-valued symbol b(x,ξ) to each smooth function with bounded derivatives b defines an injective mapping O, from the set of all such symbols to the set of all operators with smooth orbit under the canonical action of the Heisenberg group on the algebra of all adjointable operators on the Hilbert module E. It is known that O is surjective if A …
For unital $C^*$-algebras $A$ and $B$, we completely characterize the isometric ($*$-) automorphisms of their Banach space projective tensor product $A\otimes^γB$. This leads to the characterization of inner and outer isometric $*$-automorphisms of $A\otimes^γB$, as well. As an application, we provide a partial affirmative answer to a question posed by Kaijser and Sinclair, viz., we prove that for unital $C^*$-algebras $A$ and $B$, the set of norm-one unitaries of $A\otimes^γB$ coincides with $U(A) \otimes U(B)$, where $U(A)$ is the unitary group of $A$. We also establish the fact that the rel…
It is known that every semigroup of normal completely positive maps $P = {P_t: t\geq 0}$ of $B(H)$, satisfying $P_t(1) = 1$ for every $t\geq 0$, has a minimal dilation to an E_0-semigroup acting on $B(K)$ for some Hilbert space K containing H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the index of P agrees with the index of its minimal dilation to an E_0-semigroup. However, no examples were discussed, and no computations were made. In this paper we calculate the inde…
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 shall prove the following Stinespring-type theorem: there exists a triple $(π,\mathcal{H},\mathbf{V})$ associated with an unital completely positive map $Φ:\mathfrak{A}\rightarrow \mathfrak{A}$ on C* algebra $\mathfrak{A}$ with unit, where $\mathcal{H}$ is a Hilbert space, $π:\mathfrak{A\rightarrow B}(\mathcal{H})$ is a faithful representation and $\mathbf{V}$ is a linear isometry on $\mathcal{H}$ such that $π(Φ(a)=\mathbf{V}^*π(a)\mathbf{V}$ for all $a$ belong to $\mathfrak{A}$. The Nagy dilation theorem, applied to isometry $\mathbf{V}$, allows to construct a dilation of ucp-map, $Φ$, in …
In this work, we investigate the existence of a factorization for a unital completely positive map, between non-commutative probability space which do not change the expectation values of the events. These maps are called in literature stochastic maps. Using the Stinespring representations of completely positive map and assuming the existence of anti-unitary operator on Hilbert space related to these representations which satisfying some modular relations, we prove that stochastic maps with adjoint, admits a factorization.
We describe how self-adjoint ordered operator spaces, also called non-unital operator systems in the literature, can be understood as $*$-vector spaces equipped with a matrix gauge structure. We explain how this perspective has several advantages over other notions of non-unital operator systems in the literature. In particular, the category of matrix gauge $*$-vector spaces includes injective objects and a Webster-Winkler-type duality theorem, both of which we show generally fail with other notions of non-unital operator systems. As applications, we characterize those subspaces of operator sy…
Given a completely positive map, we introduce a set of algebras that we refer to as its generalized multiplicative domains. These algebras are generalizations of the traditional multiplicative domain of a completely positive map and we derive a characterization of them in the unital, trace-preserving case, in other words the case of unital quantum channels, that extends Choi's characterization of the multiplicative domains of unital maps. We also derive a characterization that is in the same flavour as a well-known characterization of bimodules, and we use these algebras to provide a new repre…
Let $τ$ be a linear map from a unital $C^*$-algebra $\CMcal A$ to a von Neumann algebra $\mathematical B$ and let $\CMcal C$ be a unital $C^*$-algebra. A map $T$ from a Hilbert $\CMcal A$-module $E$ to a von Neumann $\CMcal C$-$\CMcal B$ module $F$ is called a $τ$-map if $$\langle T(x),T(y)\rangle=τ(\langle x, y\rangle)~\mbox{for all}~x,y\in E.$$ A Stinespring type theorem for $τ$-maps and its covariant version are obtained when $τ$ is completely positive. We show that there is a bijective correspondence between the set of all $τ$-maps from $E$ to $F$ which are $(u',u)$-covariant with respect …
We generalize the main theorem of Rieffel for Morita equivalence of W*-algebras to the case of unital dual operator algebras: two unital dual operator algebras A and B have completely isometric normal representations alpha, beta such that alpha(A) is the w*-closed span of M*beta(B)M and beta(B) is the w*-closed span of Malpha(A)M* for a ternary ring of operators M (i.e. a linear space M such that MM*M \subset M if and only if there exists an equivalence functor $F:_{A}M\to_{B}M$ which "extends" to a *-functor implementing an equivalence between the categories $_{A}DM$ and $_{B}DM.$ By $_{A}M$ …
We characterize weak* closed unital vector spaces of operators on a Hilbert space $H$. More precisely, we first show that an operator system, which is the dual of an operator space, can be represented completely isometrically and weak* homeomorphically as a weak* closed operator subsystem of $B(H)$. An analogous result is proved for unital operator spaces. Finally, we give some somewhat surprising examples of dual unital operator spaces.
This article is to give an infinite dimensional analogue of a result of Choi and Effros. We say that an (not necessarily unital) operator system $T$ is \emph{dualizable} if one can find an equivalent dual matrix norm on the dual space $T^*$ such that under this dual matrix norm and the canonical dual matrix cone, $T^*$ becomes a dual operator system. We show that "a complete" operator system $T$ is dualizable if and only if $M_\infty(T)^\mathrm{sa}$ satisfies a bounded decomposition property. In this case, $$\|f\|^\mathrm{d}:= \sup \big\{\big\|[f_{i,j}(x_{k,l})]\big\|: x\in M_n(T)^+; \|x\|\leq…
