Stinespring factorization theorem

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

Formulations

It is a C * - algebra with identity and a completely positive operator in the algebra of continuous linear operators on a Hilbert space. Then there is a Hilbert space, a Hilbert space representation and a steady, linear operator, so that

Is particular.

Applies even, we can also assume that and set up the structure so that

Holds, where is the orthogonal projection on and stands for the restriction to the subspace.

Has the C * - algebra no identity, then you can adjoin one and continue with the definition of a completely positive operator and then apply above theorem. However, there may be increased by the standard.

The set of Naimark

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

It is a commutative C * - algebra with identity and a positive operator in the algebra of continuous linear operators on a Hilbert space. Then there is a Hilbert space, a Hilbert space representation and a steady, linear operator, so that

Holds, where is the orthogonal projection on and stands for the restriction to the subspace.

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

The set of Kasparov - Stinespring

The following version of the theorem of Stinespring goes back to GG Kasparov.

There were a separable and a σ - unital C * - algebra. is a completely positive operator with norm in the stable multiplier algebra. Then there is a * - homomorphism into the algebra of matrices such that:

In this case, the construction can be so set up so that the compression of the homomorphism * is the upper left corner of a matrix.