Manuscript En05-C@1 · En05-C.tex · proposition · aa213ea6

Proposition: State–effect compatibility

Stage H: a constructed bounded-sequence observable limit › Compatible states and effects · lines 1929–1952 · proof included or linked

En05-C@1#prop:limitstates, En05-C@1#eq:limitstate

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For states $ω_{n}$ on $M_{n}$ ,

ω_{\infty} ([a_{n}]) = λ ((ω_{n} (a_{n}))_{n})

is a state on $M_{\infty}$ . If $ω_{n} (a_{n}) \to p$ , then $ω_{\infty} ([a_{n}]) = p$ . For a complete finite readout with convergent outcome laws, the limit state reproduces the entire limiting law.

Proof. Norm-null representatives have scalar evaluations tending to zero, so (eq:limitstate) is well defined. Positivity and unit normalization hold coordinatewise and then under the positive functional $λ$ . On convergent sequences, $λ$ is the ordinary limit. Apply that statement to every effect of the finite readout.

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En05-C@1#eq:limitstate · line 1932
$ω_{\infty} ([a_{n}]) = λ ((ω_{n} (a_{n}))_{n})$

Proposition: State–effect compatibility

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