Questions tagged [complete-positivity]

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Example of a unital contractive map that is not completely positive on an operator system

I am aware of maps that are positive but not completely positive (for example transpose map). BUT I can not think of an example of the following type. Does there exist an operator $T$ such that a map $...
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Stinespring's theorem: can we choose the dilation to be an isometry?

Let $A$ be a $C^*$-algebra and $\varphi: A \to B(H)$ be a completely positive contractive map. Stinespring's theorem says that there exists a $*$-representation $\pi: A \to B(H')$ and a bounded ...
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"Dual" of a CP map

Let $M,N$ be von Neumann algebras, and let $\phi:M\rightarrow N$ be a normal completely positive map. I am interested in conditions when there is a "dual" normal completely positive map $\...
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Completely positive maps on Coxeter groups - the general case

In the reference below Bozejko and Speicher showed the following (for the full statement see the remark on page 9): Let $(W,S)$ be a Coxeter system, let $\mathcal{H}$ be a Hilbert space and denote ...
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7 votes
1 answer
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Can the intersection of a unitary and an irreducibly represented injective $C^*$-algebra be $\{0\}$?

Let $\mathcal{A}$ be an injective $C^*$-algebra irreducibly acting on a Hilbert space $\mathcal{H}$, and let $\phi$ be a completely positive idempotent from $\mathbb{B}(\mathcal{H})$ onto $\mathcal{A}$...
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Completely positive, unital maps acting on unitary operators [solved]

Call a completely positive, irreducible, unital map $E$ on the operators on a finite-dimensional Hilbert-space primitive if there is only a single eigenvalue with modulus 1 (all others have modulus &...
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Complete positivity with infinite dimensional auxillary spaces

The usual definition of complete positivity (e.g. Stinespring (1955), or Holevo's Statistical Structure of Quantum Theory) is that a linear map between (sub $C^*$ algebras of) the bounded operators on ...
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Maps which are both completely positive and positive

Definition:A linear map $f:\mathbb C^n\to \mathbb C^n$ is called positive if $\langle fa,a\rangle\ge0$ for all $a\in \mathbb C^n$. Equivalently, $f\in M_{n}(\mathbb C)$ is positive if it can be ...
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Reference on completely positive maps which are isometries

Let $\Phi:\mathcal{L}(H)\rightarrow \mathcal{L}(K)$ be a completely positive map sending positive self-adoint operators on a finite-dimensional Hilbert space $H$ to positive self-adoint operators on a ...
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Completely Positive Maps and their dual in Separable Hilbert Space

Consider a separable Hilbert space $\mathcal{H}$, we can first define $T(\mathcal{H})$ the trace class of $\mathcal{H}$, then $D(\mathcal{H})$ to denote the set of positive operator with trace less ...
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Bounded operators on the Stinespring representation space

Let $A$ be a $C^*$-algebra and let $\phi:A\to B(H)$ be a completely positive map. The Stinespring representation theorem constructs a representation of $A$ on a Hilbert space $K$, which is constructed ...
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Extensions of completely positive maps

It is known that for a completely bounded map $\psi:A\to B(H)$ there exist completely positive maps $\phi_1,\phi_2:A\to B(H)$ such that $$\Vert \phi_i\Vert_{cb}=\Vert \psi\Vert_{cb},$$ and the map $\...
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Completely positive maps with commuting ranges can be extended to maximal tensor product

I'm trying to do the following exercise from Brown and Ozawa's book. Exercise $3.5.1.$ Let $\varphi: A \to B(H)$ and $\psi: B \to B(H)$ be c.p. maps with commuting ranges. Show that there exists ...
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Does the Cauchy–Schwarz inequality imply 2-positivity?

Recall the following generalisation of Cauchy–Schwarz. Theorem. Let $f\colon \mathscr{A} \to \mathscr{B}$ be a linear 2-positive map between C$^*$-algebras. Then for all $a,b \in \mathscr{A}$ we ...
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Choi type matrix condition for completely positivity on a certain operator system spanned by some unitaries

Let $B_1$ and $B_2$ be $C^*$-algebras. Let $U_1, \ldots, U_n$ be some unitaries in $B_1.$ We consider the operator system $S$ spanned by $U_iU_j^*.$ Let $\phi: S \rightarrow B_2.$ Given that the ...
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Stinespring's dilation without $C^{\ast}$-algebras

Does Stinespring's dilation theorem hold if the algebra of interest is a topological $\ast$-algebra instead of the usual $C^{\ast}$-algebra? I will now state the version of Stinespring's dilation ...
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When are completely positive maps monic/epic?

In the category of C*-algebras and $*$-homomorphisms, a morphism is monic precisely when it is injective, and epic precisely when it is surjective (see Mono- and epi-morphisms for C*-algebras). Is ...
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2 votes
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Arveson index and curvature

Can someone explain me what is the intuitive idea behind Arveson Index and curvature of $E_0$ semigroups. I was reading the standard paper of Arveson, but is lost and yet to get intuition about it. An ...
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Extensions of completely positive mappings

I would like to ask the following two questions. Let $1_{\mathcal{H}}\in \mathcal{A}\subset\mathcal{B}\subset\overline{\mathcal{A}}^{SOT}\subset\mathbb{B}(\mathcal{H})$ be a sequence of $C^{\ast}$-...
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2 votes
1 answer
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Positive definite functions on G from Hilbert space vectors?

Let $G$ be a countable discrete group. Given a vector $\xi \in l^{2}(G)$, is there any way to naturally construct a positive definite function on $G$ using $\xi$? This question is rather vague and ...
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7 answers
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Positive but not completely positive?

The only example I know of a positive map which is not completely positive is the transpose map on $M_n(\mathbb{C})$. Of course, one can come up with minor perturbations of this (compose it with, or ...
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7 votes
2 answers
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When is this map completely positive?

Consider the complex $n$-by-$n$ matrices $M_n$. Suppose that $A_i$, for $i=1,\ldots,n^2$, satisfy $\mathrm{Tr}(A_i^* A_j)=\delta_{ij}$, so that together they form an orthonormal basis for $M_n$. ...
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