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Manuscript Hawk07-B@1 · Hawk07-B.tex · proposition · b9785538

Proposition: No commuting duplicate

Complete interfaces and their information tasks · lines 432–448 · proof included or linked

Hawk07-B@1#prop:noduplicate

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For $d \geq 2$ , two commuting star representations $π_{1}, π_{2}$ cannot both satisfy

π_{j} (X) V = V X (X \in B (L)) .

Proof. Commutation would give

V X Y = π_{1} (X) π_{2} (Y) V = π_{2} (Y) π_{1} (X) V = V Y X .

Injectivity of $V$ would make $B (L)$ commutative.

Displayed formulas in this passage

unlabeled · line 436
$π_{j} (X) V = V X (X \in B (L)) .$
unlabeled · line 443
$V X Y = π_{1} (X) π_{2} (Y) V = π_{2} (Y) π_{1} (X) V = V Y X .$

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