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 which satisfies a stronger, more robust condition.
Definition
Let
and
be C*-algebras. A linear map
is called a positive map if
maps positive elements to positive elements:
.
Any linear map
induces another map
![{\displaystyle {\textrm {id}}\otimes \phi :\mathbb {C} ^{k\times k}\otimes A\to \mathbb {C} ^{k\times k}\otimes B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d266d1bdf91db9652954fd9d179e872d6bfbb473)
in a natural way. If
is identified with the C*-algebra
of
-matrices with entries in
, then
acts as
![{\displaystyle {\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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42d5fb7f0c65a840641bbfd8272e59e329e9ecc4)
is called k-positive if
is a positive map and completely positive if
is k-positive for all k.
Properties
- Positive maps are monotone, i.e.
for all self-adjoint elements
.
- Since
for all self-adjoint elements
, every positive map is automatically continuous with respect to the C*-norms and its operator norm equals
. A similar statement with approximate units holds for non-unital algebras.
- The set of positive functionals
is the dual cone of the cone of positive elements of
.
Examples
- Every *-homomorphism is completely positive.[1]
- For every linear operator
between Hilbert spaces, the map
is completely positive.[2] Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
- Every positive functional
(in particular every state) is automatically completely positive.
- Given the algebras
and
of complex-valued continuous functions on compact Hausdorff spaces
, every positive map
is completely positive.
- The transposition of matrices is a standard example of a positive map that fails to be 2-positive. Let T denote this map on
. The following is a positive matrix in
: ![{\displaystyle {\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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66f72057121ac7e5fef148c14bdd7e9bcf8b7c61)
The image of this matrix under
is ![{\displaystyle {\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}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b43553457eb728f9c263e4a8c38be151f607acf)
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.) Incidentally, a map Φ is said to be co-positive if the composition Φ
T is positive. The transposition map itself is a co-positive map.
See also
References
- ^ K. R. Davidson: C*-Algebras by Example, American Mathematical Society (1996), ISBN 0-821-80599-1, Thm. IX.4.1
- ^ 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