In operator theory, Naimark's dilation theorem is a result that characterizes positive operator valued measures. It is named after Mark Naimark from his 1943 work. It can be viewed as a consequence of the Stinespring dilation theorem.
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 ) {\displaystyle L(H)} is called an operator-valued measure if it is weakly countably additive, that is, for any disjoint sequence of Borel sets { B i } {\displaystyle \{B_{i}\}} , we have
⟨ E ( ∪ i B i ) x , y ⟩ = ∑ i ⟨ E ( B i ) x , y ⟩ {\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
B → ⟨ E ( B ) x , y ⟩ {\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 | E | = sup B ‖ E ( B ) ‖ < ∞ {\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 E ( B 1 ∩ B 2 ) = E ( B 1 ) E ( B 2 ) {\displaystyle E(B_{1}\cap B_{2})=E(B_{1})E(B_{2})} for all B 1 , B 2 {\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 Φ E : C ( X ) → L ( H ) {\displaystyle \Phi _{E}:C(X)\rightarrow L(H)} in the obvious way:
⟨ Φ E ( f ) h 1 , h 2 ⟩ = ∫ X f ( x ) ⟨ E ( d x ) h 1 , h 2 ⟩ {\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
⟨ Φ E ( f ) h , h ⟩ = ∫ X f ( x ) ⟨ E ( d x ) h , h ⟩ ≤ ‖ f ‖ ∞ ⋅ | E | . {\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 Φ E ( f ) {\displaystyle \;\Phi _{E}(f)} is a bounded operator for all f, and Φ E {\displaystyle \Phi _{E}} itself is a bounded linear map as well. The properties of Φ E {\displaystyle \Phi _{E}} are directly related to those of E:
If E is positive, then Φ E {\displaystyle \Phi _{E}} , viewed as a map between C*-algebras, is also positive.
Φ E {\displaystyle \Phi _{E}} is a homomorphism if, by definition, for all continuous f on X and h 1 , h 2 ∈ H {\displaystyle h_{1},h_{2}\in H} ,
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