Naimark's Dilation Theorem - Some Preliminary Notions

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

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

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 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 .

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

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

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 ,

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

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

Similarly, to say 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, 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.)

Read more about this topic: Naimark's Dilation Theorem

Famous quotes containing the words preliminary and/or notions:

“Religion is the state of being grasped by an ultimate concern, a concern which qualifies all other concerns as preliminary and which itself contains the answer to the question of a meaning of our life.”
—Paul Tillich (1886–1965)

“What is termed Sin is an essential element of progress. Without it the world would stagnate, or grow old, or become colourless. By its curiosity Sin increases the experience of the race. Through its intensified assertion of individualism it saves us from monotony of type. In its rejection of the current notions about morality, it is one with the higher ethics.”
—Oscar Wilde (1854–1900)