Completely positive operator

Completely positive operators are examined in the mathematical sub-area of functional analysis. They are positive , linear operators between C * algebras , in which the continuations to the matrix algebras are also positive.

Definitions

A continuous, linear mapping between two C * -algebras and is called positive if it maps positive elements to positive elements, that is, if each has the form for a . ${\displaystyle P:A\rightarrow B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle P}$ ${\displaystyle P(a^{*}a)}$ ${\displaystyle a\in A}$ ${\displaystyle b^{*}b}$ ${\displaystyle b\in B}$

For let the C * -algebra of the matrices be over . This is isomorphic to the tensor product of and the C * -algebra of the complex matrices. The figure defines figures ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle M_{n}(A)}$ ${\displaystyle n\times n}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle M_{n}(\mathbb {C} )}$ ${\displaystyle n\times n}$ ${\displaystyle P:A\rightarrow B}$

${\displaystyle P_{n}=P\otimes \mathrm {id} _{M_{n}(\mathbb {C} )}\,:M_{n}(A)\rightarrow M_{n}(B),\quad P_{n}((a_{i,j})_{i,j=1,\ldots n}):=(P(a_{i,j}))_{i,j=1,\ldots n}}$.

${\displaystyle P}$is called -positive if is positive. means completely positive, if -positive is for everyone . ${\displaystyle n}$ ${\displaystyle P_{n}}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle n}$ ${\displaystyle n\in \mathbb {N} }$

Examples

Every positive, linear operator on a commutative C * algebra is completely positive.

Every state on a C * algebra is completely positive. More generally, any positive operator from a C * algebra to a commutative C * algebra is completely positive.

All * homomorphisms are completely positive. If, more generally, is a * -homomorphism and , then defines a completely positive operator. According to Stinespring's theorem , the converse applies to completely positive operators with norm less than or equal to 1. ${\displaystyle \pi :A\rightarrow B}$ ${\displaystyle b\in B}$ ${\displaystyle P(a):=b^{*}\pi (a)b}$

The transposition on the C * algebra is a positive operator that is not entirely positive. For example is ${\displaystyle P}$ ${\displaystyle M_{2}(\mathbb {C} )}$

${\displaystyle {\begin{pmatrix}1&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&1\end{pmatrix}}\in M_{4}(\mathbb {C} )=M_{2}(M_{2}(\mathbb {C} ))}$

a positive element, however

${\displaystyle P_{2}{\big (}{\begin{pmatrix}1&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&1\end{pmatrix}}{\big )}={\begin{pmatrix}P({\begin{pmatrix}1&0\\0&0\end{pmatrix}})&P({\begin{pmatrix}0&1\\0&0\end{pmatrix}})\\P({\begin{pmatrix}0&0\\1&0\end{pmatrix}})&P({\begin{pmatrix}0&0\\0&1\end{pmatrix}})\end{pmatrix}}={\begin{pmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{pmatrix}}}$

is not positive because the determinant is equal to −1. Therefore is not 2-positive. ${\displaystyle P}$

Properties and uses

Kadison-Schwarz inequality

Let it be a 2-positive, linear mapping between C * -algebras with one element and let it be . Then the Schwarz inequality applies ${\displaystyle P:A\rightarrow B}$ ${\displaystyle P(1)=1}$

${\displaystyle P(a)^{*}P(a)\leq P(a^{*}a)}$for everyone . ${\displaystyle a\in A}$

More generally applies to a completely positive figure

${\displaystyle P(a)^{*}P(a)\leq \|P\|P(a^{*}a)}$for all , ${\displaystyle a\in A}$

what is also known as the Kadison-Schwarz inequality. If only positive, then the above inequality only applies to normal elements. ${\displaystyle P}$

Nuclear C * algebras

Nuclear C * -algebras can be characterized as follows by means of completely positive operators: A C * -algebra is nuclear if and only if the identity of point-wise norm limits is completely positive, 1-bounded operators of finite order, that is, there is a network that is completely positive Operators with and for everyone and for everyone . ${\displaystyle A}$ ${\displaystyle \mathrm {id} _{A}}$ ${\displaystyle (P_{i})_{i\in I}}$ ${\displaystyle \mathrm {dim} P_{i}(A)<\infty }$ ${\displaystyle \|P_{i}\|\leq 1}$ ${\displaystyle i\in I}$ ${\displaystyle \textstyle \lim _{i\in I}\|P_{i}(a)-a\|=0}$ ${\displaystyle a\in A}$

Choi-Effros lifting theorem

The following statement, also known as the lifting theorem of Choi-Effros, applies : Let be a nuclear C * -algebra and a completely positive operator with in the quotient algebra of the C * -algebra according to the closed, two-sided ideal . Then there is a completely positive operator with and , where is the quotient mapping. ${\displaystyle A}$ ${\displaystyle P:A\rightarrow B/J}$ ${\displaystyle \|P\|\leq 1}$ ${\displaystyle B}$ ${\displaystyle J}$ ${\displaystyle Q:A\rightarrow B}$ ${\displaystyle \|Q\|\leq 1}$ ${\displaystyle P=\pi \circ Q}$ ${\displaystyle \pi :B\rightarrow B/J}$

Individual evidence

1. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-821-80599-1 , Theorem IX.4.1
2. ↑ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , exercise May 11 , 21
3. ↑ Vern I. Paulsen: Completely Bounded Maps and Operator Algebras , Cambridge University Press (2013), ISBN 0-521-81669-6 , sentence 3.3
4. ↑ Alexander S. Holevo: Statistical Structure of Quantum Theory , Springer-Verlag 2001, ISBN 3-540-42082-7 , section 3.1.1: Completely positive maps
5. ^ B. Blackadar: K-Theory for Operator-Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 15.8.1
6. ↑ M.-D. Choi, E. Effros: The completely positive lifting problem for C * -algebras , Annals of Math. (1976), Volume 104, pages 585-609