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(L^1(G))\subset B(H)\otimes^{eh}B(H) exactly when pi is norm continuous. For the case that G is abelian, we study Gamma_pi(M(G)) as a subset of the Varopoulos algebra. We also characterise positive definite elements of the Varopoulos algebra in terms of completely positive operators.
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.
We characterize completely bounded normal Jordan $*$-homomorphisms acting on von Neumann algebras. We also characterize completely positive isometries acting on noncommutative $\mathrm{L}^p$-spaces.
A linear map $u\colon \ E\to F$ between operator spaces is called completely co-bounded if it is completely bounded as a map from $E$ to the opposite of $F$. We give several simple results about completely co-bounded Schur multipliers on $B(\ell_2)$ and the Schatten class $S_p$. We also consider Herz-Schur multipliers on groups.
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…
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 $A\times_{α^{}}G$.
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.
We prove a Jordan version of Dorofeev's boundedness theorem for completely additive measues and use it to show that every (not necessarily linear nor continuous) 2-local triple derivation on a continuous JBW*-triple is a triple derivation.
We study the existence of certain completely bounded Paley projection on the anisotropic Sobolev spaces on tori. Our result should be viewed as a generalization of a similar result obtained by Pełczy{ń}ski and Wojciechowski. By a transference method, we obtain similar results on the Sobolev spaces on quantum tori.
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…
We introduce completely bounded kernels taking values in L(A,B) where A and B are C*-algebras. We show that if B is injective such kernels have a Kolmogorov decomposition precisely when they can be scaled to be completely contractive, and that this is automatic when the index set is countable.
We investigate some extremal cases of exactness constant and completely bounded projection constant. More precisely, for an $n$-dimensional operator space $E$ we prove that $λ_{cb}(E) = \sqrt{n}$ if and only if $ex(E) = \sqrt{n}$, which is equivalent to $λ_{cb}(E) < \sqrt{n}$ if and only if $ex(E) < \sqrt{n}$.
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.
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…
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.
Given a locally compact quantum group $\mathbb G$, we study the structure of completely bounded homomorphisms $π:L^1(\mathbb G)\rightarrow\mathcal B(H)$, and the question of when they are similar to $\ast$-homomorphisms. By analogy with the cocommutative case (representations of the Fourier algebra $A(G)$), we are led to consider the associated map $π^*:L^1_\sharp(\mathbb G) \rightarrow \mathcal B(H)$ given by $π^*(ω) = π(ω^\sharp)^*$. We show that the corepresentation $V_π$ of $L^\infty(\mathbb G)$ associated to $π$ is invertible if and only if both $π$ and $π^*$ are completely bounded. Moreo…
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.
We show how the proof of the Grothendieck Theorem for jointly completely bounded bilinear forms on C*-algebras by Haagerup and Musat can be modified in such a way that the method of proof is essentially C*-algebraic. To this purpose, we use Cuntz algebras rather than type III factors. Furthermore, we show that the best constant in Blecher's inequality is strictly greater than one.
We observe that for a large class of non-amenable groups $G$, one can find bounded representations of $A(G)$ on Hilbert space which are not completely bounded. We also consider restriction algebras obtained from $A(G)$, equipped with the natural operator space structure, and ask whether such algebras can be completely isomorphic to operator algebras; partial results are obtained, using a modified notion of Helson set which takes account of operator space structure. In particular, we show that if $G$ is virtually abelian, then the restriction algebra $A_G(E)$ is completely isomorphic to an oper…
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 …
For locally compact groups G and H let A(G) denote the Fourier algebra of G and B(H) the Fourier-Stieltjes algebra of H. Any continuous piecewise affine map alpha:Y -> G (where Y is an element of the open coset ring of H) induces a completely bounded homomorphism Phi_alpha:A(G) -> B(H) by setting Phi_alpha u(.)=u(alpha(.)) on Y and Phi_alpha u=0 off of Y. We show that if G is amenable then any completely bounded homomorphism Phi:A(G) -> B(H) is of this form; and this theorem fails if G contains a discrete nonabelian free group. Our result generalises results of P.J. Cohen, B. Host and of the f…
We provide alternative proofs of two recent Grothendieck theorems for jointly completely bounded bilinear forms, originally due to Pisier and Shlyakhtenko (Invent. Math. 2002) and Haagerup and Musat (Invent. Math. 2008). Our proofs are elementary and are inspired by the so-called embezzlement states in quantum information theory. Moreover, our proofs lead to quantitative estimates.
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…
We estimate the norms of many matrix coefficients of irreducible uniformly bounded representations of SL(2, R) as completely bounded multipliers of the Fourier algebra. Our results suggest that the known inequality relating the uniformly bounded norm of a representation and the completely bounded norm of its coefficients may not be optimal.
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…
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.
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 $T:\C^2 \to \C^2$ …
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 L^p(G) be the usual L^p-space for 1 =< p =< infty and A(G) be the Fourier algebra of G. Our goal is to study, in a new abstract context, the completely bounded multipliers of A(G), which we denote M_{cb}A(G). We show that M_{cb}A(G) can be characterised as the ``invariant part'' of the space of (completely) bounded normal L^{infty}(G)-bimodule maps on B(L^2(G)), the space of bounded operator on L^2(G). In doing this we develop a function theoretic description of the normal L^{infty}(X,mu)-bimodule maps on B(L^2(X,mu)), which we denote by V^{infty}(X,μ), and nam…
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 …
The purpose of this note is to construct an example of a discrete non-abelian group $G$ and a subset $E$ of $G$, not contained in any abelian subgroup, that is a completely bounded $Λ(p)$ set for all $p<\infty ,$ but is neither a Leinert set nor a weak Sidon set.
Let $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear map on normed spaces. We say that $f$ is close-to-regular when $f^{t****s}=f^{s****t}$ and we say that $f$ is completely regular when all natural extensions are equal. In this manuscript, we have some results on the close-to-regular maps and investigate the close-to-regularity of tri-linear maps. We investigate the relation between Arens regularity of bounded bilinear maps and close-to-regularity bounded tri-linear maps. We give a simple criterion for the completely regularity of tri-linear maps. We provide a necessary and suf…
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 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.
We prove that in the setting of operator spaces the result of Davis, Figiel, Johnson and Pelczynski on factoring weakly compact operators holds accordingly. Though not related directly to the main theorem we add a remark on the description of weakly compact subsets in the dual of noncommutative vector valued ${\rm L}_1$.
Given commuting families of Hermitian matrices {A1, ..., Ak} and {B1, ...., Bk}, conditions for the existence of a completely positive map L, such that L(Aj) = Bj for j = 1, ...,k, are studied. Additional properties such as unital or / and trace preserving on the map ? are also considered. Connections of the study to dilation theory, matrix inequalities, unitary orbits, and quantum information science are mentioned.
Let $E$ be an operator space in the sense of the theory recently developed by Blecher-Paulsen and Effros-Ruan. We introduce a notion of $E$-valued non commutative $L_p$-space for $1 \leq p < \infty$ and we prove that the resulting operator space satisfies the natural properties to be expected with respect to e.g. duality and interpolation. This notion leads to the definition of a ``completely p-summing" map which is the operator space analogue of the $p$-absolutely summing maps in the sense of Pietsch-Kwapień. These notions extend the particular case $p=1$ which was previously studied by Effro…
In this paper we determine all the bijective linear maps on the space of bounded observables which preserve a fixed moment or the variance. Nonlinear versions of the corresponding results are also presented.
It is a well-known result due to E. Størmer that every positive qubit map is decomposable into a sum of a completely positive map and a completely copositive map. Here, we generalize this result to tensor squares of qubit maps. Specifically, we show that any positive tensor product of a qubit map with itself is decomposable. This solves a recent conjecture by S. Fillipov and K. Magadov. We contrast this result with examples of non-decomposable positive maps arising as the tensor product of two distinct qubit maps or as the tensor square of a decomposable map from a qubit to a ququart. To show …
We develop the notions of hypercontractivity (HC) and the log-Sobolev (LS) inequality for completely bounded norms of one-parameter semigroups of super-operators acting on matrix algebras. We prove the equivalence of the completely bounded versions of HC and LS under suitable hypotheses. We also prove a version of the Gross Lemma which allows LS at general $q$ to be deduced from LS at $q=2$.
For many completely positive maps repeated compositions will eventually become entanglement breaking. To quantify this behaviour we develop a technique based on the Schmidt number: If a completely positive map breaks the entanglement with respect to any qubit ancilla, then applying it to part of a bipartite quantum state will result in a Schmidt number bounded away from the maximum possible value. Iterating this result puts a successively decreasing upper bound on the Schmidt number arising in this way from compositions of such a map. By applying this technique to completely positive maps in d…
