Naimark's dilation theorem

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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.

Note[edit]

In the mathematical literature, one may also find other results that bear Naimark's name.

Some preliminary notions[edit]

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 L(H) is called a operator-valued measure if it is weakly countably additive, that is, for any disjoint sequence of Borel sets \{ B_i \}, we have


\langle E (\cup _i B_i) x, y \rangle = \sum_i \langle E (B_i) x, y \rangle

for all x and y. Some terminology for describing such measures are:

  • E is called regular if the scalar valued measure

B \rightarrow \langle E (B) x, y \rangle

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| = \sup_B \|E(B) \| < \infty.
  • 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 E (B_1 \cap B_2) = E(B_1) E(B_2) for all  B_1, B_2 .

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 \Phi _E : C(X) \rightarrow L(H) in the obvious way:

\langle \Phi _E (f) h_1 , h_2 \rangle = \int _X f d \langle E(B) h_1, h_2 \rangle

The boundedness of E implies, for all h of unit norm


\langle \Phi _E (f) h , h \rangle = \int _X f d \langle E(B) h, h \rangle \leq \| f \| \cdot |E| .

This shows \; \Phi _E (f) is a bounded operator for all f, and \Phi _E itself is a bounded linear map as well.

The properties of \Phi_E are directly related to those of E:

  • If E is positive, then \Phi_E, viewed as a map between C*-algebras, is also positive.
  • \Phi_E is a homomorphism if, by definition, for all continuous f on X and h_1, h_2 \in H,

\langle \Phi_E (fg) h_1, h_2 \rangle = \int _X f \cdot g \; d \langle E(B) h_1, h_2 \rangle 
= \langle \Phi_E (f) \Phi_E (g) h_1 , h_2 \rangle.

Take f and g to be indicator functions of Borel sets and we see that \Phi _E is a homomorphism if and only if E is spectral.

  • Similarly, to say \Phi_E respects the * operation means

\langle \Phi_E ( {\bar f} ) h_1, h_2 \rangle = \langle \Phi_E (f) ^* h_1 , h_2 \rangle.

The LHS is

 
\int _X {\bar f} \; d \langle E(B) h_1, h_2 \rangle,

and the RHS is

 
\langle h_1, \Phi_E (f) h_2 \rangle = \int _X {\bar f} \; d \langle E(B) h_2, h_1 \rangle

So, for all B, \langle E(B) h_1, h_2 \rangle = \langle E(B) h_2, h_1 \rangle, i.e. E(B) is self adjoint.

  • Combining the previous two facts gives the conclusion that \Phi _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 projection-valued measure or PVM.)

Naimark's theorem[edit]

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 V: K \rightarrow H, and a self-adjoint, spectral L(K)-valued measure on X, F, such that

\; E(B) = V F(B) V^*.

Proof[edit]

We now sketch the proof. The argument passes E to the induced map \Phi_E and uses Stinespring's dilation theorem. Since E is positive, so is \Phi_E as a map between C*-algebras, as explained above. Furthermore, because the domain of \Phi _E, C(X), is an abelian C*-algebra, we have that \Phi_E is completely positive. By Stinespring's result, there exists a Hilbert space K, a *-homomorphism \pi : C(X) \rightarrow L(K), and operator V: K \rightarrow H such that

\; \Phi_E(f) = V \pi (f) V^*.

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.

Finite-dimensional case[edit]

In the finite-dimensional case, there is a somewhat more explicit formulation.

Suppose now X = \{1, \cdots, n \}, therefore C(X) is the finite-dimensional algebra \mathbb{C}^n, and H has finite dimension m. A positive operator-valued 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 \; \sum _i E_i = I where I is the identity operator. (See the article on POVM for relevant applications.) This would mean the induced map \Phi _E is unital. It can be assumed with no loss of generality that each E_i is a rank-one projection onto some x_i \in \mathbb{C}^m. 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 \sum _{i=1}^n x_i x_i^* = I 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 non-square matrix

 M = \begin{bmatrix} x_1 & \cdots x_n \end{bmatrix}

is an isometry, i.e. M M^* = I. If we can find a (n-m) \times n matrix N where

U = \begin{bmatrix} M \\ N \end{bmatrix}

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[edit]

  • V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2003.