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

Proposition: Product-preserving observable limits

Stage 1: three different locality questions › The actual observable-limit condition · lines 502–525 · proof included or linked

Oai280-A@1#prop:observablelimit

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Suppose bounded generators $a_{n}, b_{n}$ commute at every sufficiently fine scale. Suppose comparison maps into a common $C^{*}$ -algebra satisfy

j_{n} (a_{n}) \to a, j_{n} (b_{n}) \to b,

j_{n} (a_{n} b_{n}) - j_{n} (a_{n}) j_{n} (b_{n}) \to 0, j_{n} (b_{n} a_{n}) - j_{n} (b_{n}) j_{n} (a_{n}) \to 0

in norm. Then $[a, b] = 0$ . Two local algebras generated by such mutually commuting limits commute.

Proof. Subtract the two cross-product comparisons and use $a_{n} b_{n} = b_{n} a_{n}$ . The limiting commutator is zero. Finite polynomials in the two generator families commute, and norm closure preserves this equality.

Displayed formulas in this passage

unlabeled · line 507
$j_{n} (a_{n}) \to a, j_{n} (b_{n}) \to b,$
unlabeled · line 510
$j_{n} (a_{n} b_{n}) - j_{n} (a_{n}) j_{n} (b_{n}) \to 0, j_{n} (b_{n} a_{n}) - j_{n} (b_{n}) j_{n} (a_{n}) \to 0$

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section: The actual observable-limit condition