Naimark's dilation theorem

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

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

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

• E is called regular if the scalar valued measure

${\displaystyle B\rightarrow \langle E(B)x,y\rangle }$

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 ${\displaystyle |E|=\sup _{B}\|E(B)\|<\infty }$.
• 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 ${\displaystyle E(B_{1}\cap B_{2})=E(B_{1})E(B_{2})}$ for all ${\displaystyle B_{1},B_{2}}$.

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

${\displaystyle \langle \Phi _{E}(f)h_{1},h_{2}\rangle =\int _{X}f(x)\langle E(dx)h_{1},h_{2}\rangle }$

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

${\displaystyle \langle \Phi _{E}(f)h,h\rangle =\int _{X}f(x)\langle E(dx)h,h\rangle \leq \|f\|_{\infty }\cdot |E|.}$

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

The properties of ${\displaystyle \Phi _{E}}$ are directly related to those of E:

• If E is positive, then ${\displaystyle \Phi _{E}}$, viewed as a map between C*-algebras, is also positive.
• ${\displaystyle \Phi _{E}}$ is a homomorphism if, by definition, for all continuous f on X and ${\displaystyle h_{1},h_{2}\in H}$,

${\displaystyle \langle \Phi _{E}(fg)h_{1},h_{2}\rangle =\int _{X}f(x)\cdot g(x)\;\langle E(dx)h_{1},h_{2}\rangle =\langle \Phi _{E}(f)\Phi _{E}(g)h_{1},h_{2}\rangle .}$

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

• Similarly, to say ${\displaystyle \Phi _{E}}$ respects the * operation means

${\displaystyle \langle \Phi _{E}({\bar {f}})h_{1},h_{2}\rangle =\langle \Phi _{E}(f)^{*}h_{1},h_{2}\rangle .}$

The LHS is

${\displaystyle \int _{X}{\bar {f}}\;\langle E(dx)h_{1},h_{2}\rangle ,}$

and the RHS is

${\displaystyle \langle h_{1},\Phi _{E}(f)h_{2}\rangle ={\overline {\langle \Phi _{E}(f)h_{2},h_{1}\rangle }}=\int _{X}{\bar {f}}(x)\;{\overline {\langle E(dx)h_{2},h_{1}\rangle }}=\int _{X}{\bar {f}}(x)\;\langle h_{1},E(dx)h_{2}\rangle }$

So, taking f a sequence of continuous functions increasing to the indicator function of B, we get ${\displaystyle \langle E(B)h_{1},h_{2}\rangle =\langle h_{1},E(B)h_{2}\rangle }$, i.e. E(B) is self adjoint.

• Combining the previous two facts gives the conclusion that ${\displaystyle \Phi _{E}}$ 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.)

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

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

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

Suppose now ${\displaystyle X=\{1,\dotsc ,n\}}$, therefore C(X) is the finite-dimensional algebra ${\displaystyle \mathbb {C} ^{n}}$, and H has finite dimension m. A positive operator-valued measure E then assigns each i a positive semidefinite m × m matrix ${\displaystyle E_{i}}$. Naimark's theorem now states that there is a projection-valued measure on X whose restriction is E.

Of particular interest is the special case when ${\displaystyle \sum _{i}E_{i}=I}$ where I is the identity operator. (See the article on POVM for relevant applications.) In this case, the induced map ${\displaystyle \Phi _{E}}$ is unital. It can be assumed with no loss of generality that each ${\displaystyle E_{i}}$ takes the form ${\displaystyle x_{i}x_{i}^{*}}$ for some potentially subnorrmalized vector ${\displaystyle x_{i}\in \mathbb {C} ^{m}}$. Under such assumptions, the case ${\displaystyle n<m}$ is excluded and we must have either

1. ${\displaystyle n=m}$ and E is already a projection-valued measure (because ${\displaystyle \sum _{i=1}^{n}x_{i}x_{i}^{*}=I}$ if and only if ${\displaystyle \{x_{i}\}}$ is an orthonormal basis),
2. ${\displaystyle n>m}$ and ${\displaystyle \{E_{i}\}}$ does not consist of mutually orthogonal projections.

For the second possibility, the problem of finding a suitable projection-valued measure now becomes the following problem. By assumption, the non-square matrix

${\displaystyle M={\begin{bmatrix}x_{1}\cdots x_{n}\end{bmatrix}}}$

is a co-isometry, that is ${\displaystyle MM^{*}=I}$. If we can find a ${\displaystyle (n-m)\times n}$ matrix N where

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

is a n × n unitary matrix, the projection-valued measure 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.

In the physics literature, it is common to see the spelling “Neumark” instead of “Naimark.” The latter variant is according to the romanization of Russian used in translation of Soviet journals, with diacritics omitted (originally Naĭmark). The former is according to the etymology of the surname of Mark Naimark.

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