Manuscript Oai280-A@1 · Oai280-A.tex · lemma · 27038823

Lemma: Equal-snapshot separation

A tensor-free comparison used in the finite certificate · lines 610–654 · proof included or linked

Oai280-A@1#lem:snapshot, Oai280-A@1#eq:quantitativesnapshot

For a unital $C^{*}$ -subalgebra $A \subset M$ and $h = h^{*} \in M$ , the following are equivalent:

h \in A; ω (h) = η (h) whenever ω ∣_{A} = η ∣_{A} .

Moreover,

ω ∣_{A} = η ∣_{A} sup ∣ ω (h) - η (h) ∣ = 2 dist (h, A_{sa}) .

The supremum is over states of $M$ .

Proof. For $a = a^{*} \in A$ ,

∣ ω (h) - η (h) ∣ = ∣ ω (h - a) - η (h - a) ∣ \leq 2∥ h - a ∥.

This proves the upper bound.

If the distance is positive, real Hahn–Banach separation on $M_{sa}$ gives a norm-one real functional annihilating $A_{sa}$ and taking value $dist (h, A_{sa})$ on $h$ . Its complexification is a bounded Hermitian functional $f$ of the same norm. Write its Jordan decomposition $f = f_{+} - f_{-}$ . Because $1 \in A$ ,

f_{+} (1) = f_{-} (1) = 1/2.

Thus $2 f_{+}$ and $2 f_{-}$ are states with equal restrictions to $A$ , and their difference on $h$ is twice the distance. If the distance is zero, the upper bound proves the equality.

Jordan decomposition is the standard $C^{*}$ -algebraic background used here Pedersen.

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Oai280-A@1#eq:quantitativesnapshot · line 620
$ω ∣_{A} = η ∣_{A} sup ∣ ω (h) - η (h) ∣ = 2 dist (h, A_{sa}) .$
unlabeled · line 614
$h \in A; ω (h) = η (h) whenever ω ∣_{A} = η ∣_{A} .$
unlabeled · line 631
$∣ ω (h) - η (h) ∣ = ∣ ω (h - a) - η (h - a) ∣ \leq 2∥ h - a ∥.$
unlabeled · line 645
$f_{+} (1) = f_{-} (1) = 1/2.$

Lemma: Equal-snapshot separation

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