All Questions
Tagged with oa.operator-algebras complete-positivity
16 questions
3
votes
0
answers
77
views
+50
Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps
Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function
$$
m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)},
$$
where $\lambda_{\max}$ denotes the largest eigenvalue....
- 673
18
votes
7
answers
4k
views
What are known examples of positive but not completely positive maps?
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 ...
- 342
7
votes
2
answers
2k
views
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
$M_n$. ...
- 4,713
2
votes
0
answers
267
views
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 $...
- 231
3
votes
0
answers
179
views
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 ...
- 175
4
votes
0
answers
131
views
"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 $\...
- 18.7k
7
votes
1
answer
264
views
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 $\mathcal{A}$...
- 10.1k
3
votes
0
answers
181
views
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 &...
- 31
8
votes
1
answer
547
views
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 ...
- 2,689
3
votes
1
answer
325
views
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 ...
- 3,984
1
vote
1
answer
190
views
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 ...
- 18.7k
1
vote
1
answer
441
views
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 $\...
- 18.7k
2
votes
1
answer
122
views
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 ...
- 2,753
5
votes
1
answer
516
views
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 ...
- 8,098
2
votes
1
answer
222
views
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 ...
- 1,411
4
votes
0
answers
282
views
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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