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

$\langle E(\cup _{i}B_{i})x,y\rangle =\sum _{i}\langle E(B_{i})x,y\rangle$

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 $E(B_{1}\cap B_{2})=E(B_{1})E(B_{2})$.

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 $\Phi _{E}:C(X)\rightarrow L(H)$ in the obvious way:

$\langle \Phi _{E}(f)h_{1},h_{2}\rangle =\int _{X}fd\langle E(B)h_{1},h_{2}\rangle$

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

$\langle \Phi _{E}(f)h,h\rangle =\int _{X}fd\langle E(B)h,h\rangle \leq \|f\|\cdot |E|.$

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

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

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

Proof

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

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. $MM^{*}=I$. 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

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