completely positive map

{{Short description|C*-algebra mapping preserving positive elements}}

In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one that satisfies a stronger, more robust condition.

Definition

Let $A$ and $B$ be C*-algebras. A linear map $\phi: A\to B$ is called a positive map if $\phi$ maps positive elements to positive elements: $a\geq 0 \implies \phi(a)\geq 0$ .

Any linear map $\phi:A\to B$ induces another map

: $\textrm{id} \otimes \phi : \mathbb{C}^{k \times k} \otimes A \to \mathbb{C}^{k \times k} \otimes B$

in a natural way. If $\mathbb{C}^{k\times k}\otimes A$ is identified with the C*-algebra $A^{k\times k}$ of $k\times k$ -matrices with entries in $A$ , then $\textrm{id}\otimes\phi$ acts as

: $\begin{pmatrix}
a_{11} & \cdots & a_{1k} \\
\vdots & \ddots & \vdots \\
a_{k1} & \cdots & a_{kk}
\end{pmatrix} \mapsto \begin{pmatrix}
\phi(a_{11}) & \cdots & \phi(a_{1k}) \\
\vdots & \ddots & \vdots \\
\phi(a_{k1}) & \cdots & \phi(a_{kk})
\end{pmatrix}.$

We then say $\phi$ is k-positive if $\textrm{id}_{\mathbb{C}^{k\times k}} \otimes \phi$ is a positive map and completely positive if $\phi$ is k-positive for all k.

Properties

Positive maps are monotone, i.e. $a_1\leq a_2\implies \phi(a_1)\leq\phi(a_2)$ for all self-adjoint elements $a_1,a_2\in A_{sa}$ .
Since $-\|a\|_A 1_A \leq a \leq \|a\|_A 1_A$ for all self-adjoint elements $a\in A_{sa}$ , every positive map is automatically continuous with respect to the C*-norms and its operator norm equals $\|\phi(1_A)\|_B$ . A similar statement with approximate units holds for non-unital algebras.
The set of positive functionals $\to\mathbb{C}$ is the dual cone of the cone of positive elements of $A$ .

Examples

Every *-homomorphism is completely positive.K. R. Davidson: C*-Algebras by Example, American Mathematical Society (1996), ISBN 0-821-80599-1, Thm. IX.4.1
For every linear operator $V:H_1\to H_2$ between Hilbert spaces, the map $L(H_1)\to L(H_2), \ A \mapsto V A V^\ast$ is completely positive.R.V. Kadison, J. R. Ringrose: Fundamentals of the Theory of Operator Algebras II, Academic Press (1983), ISBN 0-1239-3302-1, Sect. 11.5.21 Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
Every positive functional $\phi:A \to \mathbb{C}$ (in particular every state) is automatically completely positive.
Given the algebras $C(X)$ and $C(Y)$ of complex-valued continuous functions on compact Hausdorff spaces $X, Y$ , every positive map $C(X)\to C(Y)$ is completely positive.
The transposition of matrices is a standard example of a positive map that fails to be 2-positive. Let {{math|T}} denote this map on $\mathbb{C}^{n \times n}$ . The following is a positive matrix in $\mathbb{C}^{2\times 2} \otimes \mathbb{C}^{2\times 2}$ :

\begin{bmatrix} \begin{pmatrix}1&0\\0&0\end{pmatrix}& \begin{pmatrix}0&1\\0&0\end{pmatrix}\\ \begin{pmatrix}0&0\\1&0\end{pmatrix}& \begin{pmatrix}0&0\\0&1\end{pmatrix} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ \end{bmatrix}. The image of this matrix under

I_2 \otimes T

\begin{bmatrix}
\begin{pmatrix}1&0\\0&0\end{pmatrix}^T&
\begin{pmatrix}0&1\\0&0\end{pmatrix}^T\\
\begin{pmatrix}0&0\\1&0\end{pmatrix}^T&
\begin{pmatrix}0&0\\0&1\end{pmatrix}^T
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix} ,

which is clearly not positive, having determinant −1. Moreover, the eigenvalues of this matrix are 1,1,1 and −1. (This matrix happens to be the Choi matrix of T, in fact.) {{pb}} Incidentally, a map Φ is said to be co-positive if the composition Φ

\circ

T is positive. The transposition map itself is a co-positive map.

References

Category:C*-algebras

completely positive map

Definition

Properties

Examples

See also

References