Fine topology (potential theory)

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

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 $L(H)$ 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

$⟨E({\cup }_i{B}_i)x,y⟩=\sum _i⟨E(B_i)x,y⟩$

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

• E is called regular if the scalar valued measure

$B\to ⟨E(B)x,y⟩$

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 $\left|E\right|={\sup }_B\Vert E(B)\Vert <\mathrm{\infty }$.

• 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 $E(B_1\cap B_2)=E(B_1)E(B_2)$.

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 ${\mathrm{\Phi }}_E:C(X)\to L(H)$ in the obvious way:

$⟨{\mathrm{\Phi }}_E(f)h_1,h_2⟩=\underset{X}{\int }fd⟨E(B)h_1,h_2⟩$

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

$⟨{\mathrm{\Phi }}_E(f)h,h⟩=\underset{X}{\int }fd⟨E(B)h,h⟩\le \Vert f\Vert \cdot \left|E\right|.$

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

The properties of ${\mathrm{\Phi }}_E$ are directly related to those of E:

• If E is positive, then ${\mathrm{\Phi }}_E$, viewed as a map between C*-algebras, is also positive.

• ${\mathrm{\Phi }}_E$ is a homomorphism if, by definition, for all continuous f on X and $h_1,h_2\in H$,

$⟨{\mathrm{\Phi }}_E(fg)h_1,h_2⟩=\underset{X}{\int }f\cdot gd⟨E(B)h_1,h_2⟩=⟨{\mathrm{\Phi }}_E(f){\mathrm{\Phi }}_E(g)h_1,h_2⟩.$

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

• Similarly, to say ${\mathrm{\Phi }}_E$ respects the * operation means

$⟨{\mathrm{\Phi }}_E(\overline{f})h_1,h_2⟩=⟨{\mathrm{\Phi }}_E(f{)}^{*}{h}_1,h_2⟩.$

The LHS is

$\underset{X}{\int }\overline{f}d⟨E(B)h_1,h_2⟩,$

and the RHS is

$⟨h_1,{\mathrm{\Phi }}_E(f)h_2⟩=\underset{X}{\int }\overline{f}d⟨E(B)h_2,h_1⟩$

So, for all B, $⟨E(B)h_1,h_2⟩=⟨E(B)h_2,h_1⟩$, i.e. E(B) is self adjoint.

• Combining the previous two facts gives the conclusion that ${\mathrm{\Phi }}_E$ 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

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

$E(B)=VF(B)V^{*}.$

Proof

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

${\mathrm{\Phi }}_E(f)=V\pi (f)V^{*}.$

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

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 ${\mathrm{\Phi }}_E$ 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

$M=\left[\begin{array}{cc}{x}_1& \cdots x_n\end{array}\right]$

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

$U=\left[\begin{array}{c}M\\ N\end{array}\right]$

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.