 Naimark's dilation theorem

In operator theory, Naimark's dilation theorem is a result that characterizes positive operator valued measures. It can be viewed as a consequence of Stinespring's dilation theorem.
Contents
Note
In the mathematical literature, one may also find other results that bear Naimark's name.
Some preliminary notions
Let X be a compact Hausdorff space, H be a Hilbert space, and L(H) the Banach space of bounded operators on H. A mapping E from the Borel σalgebra on X to L(H) is called a operatorvalued measure if it is weakly countably additive, that is, for any disjoint sequence of Borel sets {B_{i}}, we have
for all x and y. Some terminology for describing such measures are:
 E is called regular if the scalar valued measure
is a regular Borel measure, meaning all compact sets have finite total variation and the measure of a set can be approximated by those of open sets.
 E is called bounded if .
 E is called positive if E(B) is a positive operator for all B.
 E is called selfadjoint if E(B) is selfadjoint for all B.
 E is called spectral if .
We will assume throughout that E is regular.
Let C(X) denote the abelian C*algebra of continuous functions on X. If E is regular and bounded, it induces a map in the obvious way:
The boundedness of E implies, for all h of unit norm
This shows is a bounded operator for all f, and Φ_{E} itself is a bounded linear map as well.
The properties of Φ_{E} are directly related to those of E:
 If E is positive, then Φ_{E}, viewed as a map between C*algebras, is also positive.
 Φ_{E} is a homomorphism if, by definition, for all continuous f on X and ,
Take f and g to be indicator functions of Borel sets and we see that Φ_{E} is a homomorphism if and only if E is spectral.
 Similarly, to say Φ_{E} respects the * operation means
The LHS is
and the RHS is
So, for all B, , i.e. E(B) is self adjoint.
 Combining the previous two facts gives the conclusion that Φ_{E} is a *homomorphism if and only if E is spectral and self adjoint. (When E is spectral and self adjoint, E is said to be a projectionvalued measure or PVM.)
Naimark's theorem
The theorem reads as follows: Let E be a positive L(H)valued measure on X. There exists a Hilbert space K, a bounded operator , and a selfadjoint, spectral L(K)valued measure on X, F, such that
Proof
We now sketch the proof. The argument passes E to the induced map Φ_{E} and uses Stinespring's dilation theorem. Since E is positive, so is Φ_{E} as a map between C*algebras, as explained above. Furthermore, because the domain of Φ_{E}, C(X), is an abelian C*algebra, we have that Φ_{E} is completely positive. By Stinespring's result, there exists a Hilbert space K, a *homomorphism , and operator such that
Since π is a *homomorphism, its corresponding operatorvalued measure F is spectral and self adjoint. It is easily seen that F has the desired properties.
Finite dimensional case
In the finite dimensional case, there is a somewhat more explicit formulation.
Suppose now , therefore C(X) is the finite dimensional algebra , and H has finite dimension m. A positive operatorvalued measure E then assigns each i a positive semidefinite m X m matrix E_{i}. Naimark's theorem now says there is a projection valued measure on X whose restriction is E.
Of particular interest is the special case when where I is the identity operator. (See the article on POVM for relevant applications.) This would mean the induced map Φ_{E} is unital. It can be assumed with no loss of generality that each E_{i} is a rankone projection onto some . Under such assumptions, the case n < m is excluded and we must have either:
1) n = m and E is already a projection valued measure. (Because if and only if {x_{i}} is an orthonormal basis.) ,or
2) n > m and {E_{i}} does not consist of mutually orthogonal projections.
For the second possibility, the problem of finding a suitable PVM now becomes the following: By assumption, the nonsquare matrix
is an isometry, i.e. MM ^{*} = I. If we can find a matrix N where
is a n X n unitary matrix, the PVM whose elements are projections onto the column vectors of U will then have the desired properties. In principle, such a N can always be found.
References
 V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2003.
Categories: Operator theory
 Measure theory
 Theorems in functional analysis
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