Set of Stinespring

The set of Stine Spring , named after W. Forrest Stine Spring is a set of the mathematical branch of functional analysis in 1955. He says that completely positive operators on C * -algebras essentially compressions of Hilbert space representations are.

Formulations

Let us be a C * -algebra with one element and a completely positive operator in the algebra of continuous, linear operators over a Hilbert space . Then there is a Hilbert space , a Hilbert space representation, and a continuous linear operator such that ${\displaystyle A}$ ${\displaystyle \varphi :A\rightarrow L(H)}$ ${\displaystyle H}$ ${\displaystyle {\hat {H}}}$ ${\displaystyle \pi :A\rightarrow L({\hat {H}})}$ ${\displaystyle V:H\rightarrow {\hat {H}}}$

${\displaystyle \varphi (a)=V^{*}\pi (a)V}$ for all ${\displaystyle a\in A}$

In particular is . ${\displaystyle \|\varphi \|=\|V^{*}V\|=\|\varphi (1)\|}$

If it is even true , one can also assume that and set up the construction in such a way that ${\displaystyle \varphi (1)=\mathrm {id_{H}} }$ ${\displaystyle H\subset {\hat {H}}}$

${\displaystyle \varphi (a)=P_{H}\pi (a)|_{H}}$ for all ${\displaystyle a\in A}$

holds, where the orthogonal projection is on and stands for the restriction to the subspace . ${\displaystyle P_{H}}$ ${\displaystyle H}$ ${\displaystyle |_{H}}$ ${\displaystyle H\subset {\hat {H}}}$

If the C * -algebra has no unity element, one can adjoint one and continue with the definition to a completely positive operator and apply the above sentence to it. However, this may increase the norm of . ${\displaystyle A}$ ${\displaystyle \varphi }$ ${\displaystyle \varphi (1):=\mathrm {id_{H}} }$ ${\displaystyle \varphi }$

Naimark's theorem

The set of Naimark from 1943, named after Mark Naimark , is an important precursor of the set of Stinespring, he handled the case of commutative C * -algebras:

Let it be a commutative C * -algebra with one element and a positive operator in the algebra of continuous, linear operators over a Hilbert space . Then there is a Hilbert space , a Hilbert space representation, and a continuous linear operator such that ${\displaystyle A}$ ${\displaystyle \varphi :A\rightarrow L(H)}$ ${\displaystyle H}$ ${\displaystyle {\hat {H}}\supset H}$ ${\displaystyle \pi :A\rightarrow L({\hat {H}})}$ ${\displaystyle V:H\rightarrow {\hat {H}}}$

${\displaystyle \varphi (a)=P_{H}\pi (a)|_{H}}$ for all ${\displaystyle a\in A}$

holds, where the orthogonal projection is on and stands for the restriction to the subspace . ${\displaystyle P_{H}}$ ${\displaystyle H}$ ${\displaystyle |_{H}}$ ${\displaystyle H\subset {\hat {H}}}$

This theorem follows easily from the above second version of Stinespring's theorem and the fact that positive operators on commutative C * algebras are automatically completely positive.

Kasparov-Stinespring's theorem

The following version of Stinespring's theorem goes back to GG Kasparow .

Let there be a separable and a σ-unital C * -algebra. is a fully positive operator with standard in the stable multiplier algebra over . Then there is a * homomorphism over into the algebra of the matrices such that: ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \varphi :A\rightarrow M^{s}(B)}$ ${\displaystyle \leq 1}$ ${\displaystyle B}$ ${\displaystyle \pi :A\rightarrow M_{2}(M^{s}(B))}$ ${\displaystyle 2\times 2}$ ${\displaystyle M^{s}(B)}$

${\displaystyle {\begin{pmatrix}\varphi (a)&0\\0&0\end{pmatrix}}={\begin{pmatrix}1&0\\0&0\end{pmatrix}}\cdot \pi (a)\cdot {\begin{pmatrix}1&0\\0&0\end{pmatrix}}}$for everyone . ${\displaystyle a\in A}$

In this case the construction can be set up in such a way that the compression of the * homomorphism is the upper left corner of a matrix. ${\displaystyle 2\times 2}$

Individual evidence

1. ^ W. Stinespring: Positive functions on C * -algebras , Proceedings Amer. Math. Soc. (1955), Volume 6, Pages 211-216
2. ^ NP Brown, N. Ozawa: C * -Algebras and Finite-Dimensional Approximations , American Mathematical Soc. (2008), Volume 88, ISBN 0-8218-7250-8 , Theorem 1.5.3
3. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-821-80599-1 , Theorem IX.4.3
4. ^ NP Brown, N. Ozawa: C * -Algebras and Finite-Dimensional Approximations , American Mathematical Soc. (2008), Volume 88, ISBN 0-8218-7250-8 , sentence 2.2.1
5. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-821-80599-1 , Theorem IX.4.2
6. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-821-80599-1 , Theorem IX.4.1
7. ^ GG Kasparow: Hilbert-C * modules: theorems of Stinespring and Voiculescu , Journal Operator Theory (1980), Volume 4, pages 133-150