Manuscript En03-A@1 · En03-A.tex · proposition · 5398614e

Proposition: Accessible-algebra criterion

Scale, records, and residual instruments › Accessible output algebras · lines 2617–2639 · proof included or linked

En03-A@1#prop:accessible, En03-A@1#eq:accessible-pullback, En03-A@1#eq:accessible-norm

Agreement on $O_{L}$ for every input is equivalent to

F^{*} (A) = G^{*} (A) (A \in O_{L}) .

Moreover,

d_{O_{L}} (F, G) = \frac{1}{2} A = A^{†} \in O_{L} ∥ A ∥_{\infty} \leq 1 sup ∥ F^{*} (A) - G^{*} (A) ∥_{\infty} .

Proof. Self-adjoint expectations identify the Hermitian adjoint images. Every algebra element is a complex linear combination of self-adjoint elements. Interchange the two suprema and optimize the Hermitian expectation spectrally.

Displayed formulas in this passage

En03-A@1#eq:accessible-pullback · line 2620
$F^{*} (A) = G^{*} (A) (A \in O_{L}) .$
En03-A@1#eq:accessible-norm · line 2625
$d_{O_{L}} (F, G) = \frac{1}{2} A = A^{†} \in O_{L} ∥ A ∥_{\infty} \leq 1 sup ∥ F^{*} (A) - G^{*} (A) ∥_{\infty} .$

Proposition: Accessible-algebra criterion

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