Naimark's dilation theorem
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 σ-algebra on X to is called a operator-valued measure if it is weakly countably additive, that is, for any disjoint sequence of Borel sets , 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 self-adjoint if E(B) is self-adjoint for all B.
- E is called spectral if it is self-adjoint and for all .
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 itself is a bounded linear map as well.
The properties of are directly related to those of E:
- If E is positive, then , viewed as a map between C*-algebras, is also positive.
- 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 is a homomorphism if and only if E is spectral.
- Similarly, to say respects the * operation means
The LHS is
and the RHS is
So, taking f a sequence of continuous functions increasing to the indicator function of B, we get , i.e. E(B) is self adjoint.
- Combining the previous two facts gives the conclusion that 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 projection-valued measure or PVM.)
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 self-adjoint, spectral L(K)-valued measure on X, F, such that
We now sketch the proof. The argument passes E to the induced map and uses Stinespring's dilation theorem. Since E is positive, so is as a map between C*-algebras, as explained above. Furthermore, because the domain of , C(X), is an abelian C*-algebra, we have that 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 operator-valued measure F is spectral and self adjoint. It is easily seen that F has the desired properties.
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 operator-valued measure E then assigns each i a positive semidefinite m X m matrix . 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 is unital. It can be assumed with no loss of generality that each is a rank-one projection onto some . Under such assumptions, the case is excluded and we must have either:
1) and E is already a projection valued measure. (Because if and only if is an orthonormal basis.) ,or
2) and 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 non-square matrix
is an isometry, i.e. . 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.
- V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2003.