Completely positive map

{{Short description|C*-algebra mapping preserving positive elements}} {{Use American English|date=January 2019}} {{More citations needed|date=July 2020}} 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 <math>A</math> and <math>B</math> be C*-algebras. A linear map <math>\phi: A\to B</math> is called a '''positive map''' if <math>\phi</math> maps positive elements to positive elements: <math>a\geq 0 \implies \phi(a)\geq 0</math>.

Any linear map <math>\phi:A\to B</math> induces another map

:<math>\textrm{id} \otimes \phi : \mathbb{C}^{k \times k} \otimes A \to \mathbb{C}^{k \times k} \otimes B</math>

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

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

== Properties ==

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

== Examples ==

* Every *-homomorphism is completely positive.<ref>K. R. Davidson: ''C*-Algebras by Example'', American Mathematical Society (1996), ISBN 0-821-80599-1, Thm. IX.4.1</ref> * For every linear operator <math>V:H_1\to H_2</math> between Hilbert spaces, the map <math>L(H_1)\to L(H_2), \ A \mapsto V A V^\ast</math> is completely positive.<ref>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</ref> Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps. * Every positive functional <math>\phi:A \to \mathbb{C}</math> (in particular every state) is automatically completely positive. * Given the algebras <math>C(X)</math> and <math>C(Y)</math> of complex-valued continuous functions on compact Hausdorff spaces <math>X, Y</math>, every positive map <math>C(X)\to C(Y)</math> 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 <math>\mathbb{C}^{n \times n}</math>. The following is a positive matrix in <math>\mathbb{C}^{2\times 2} \otimes \mathbb{C}^{2\times 2}</math>: <math display="block"> \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}. </math> The image of this matrix under <math>I_2 \otimes T</math> is <math display="block"> \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} , </math> 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 <math>\circ</math> ''T'' is positive. The transposition map itself is a co-positive map.

==References== {{reflist}}

Category:C*-algebras