Manuscript Hawk05-C@1 · Hawk05-C.tex · proposition · ba0719f1

Proposition: No commuting duplicate action on one code

Access algebras and independently certified recovery › The precise obstruction to duplicate logical algebras · lines 1622–1642 · proof included or linked

Hawk05-C@1#prop:duplicates

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Let $V : C \to H$ be an isometry. There cannot be two commuting unital star representations $π_{1}, π_{2} : B (C) \to B (H)$ such that

π_{1} (X) V = V X, π_{2} (X) V = V X

for every $X \in B (C)$ , when $d \geq 2$ .

Proof. For all $X, Y$ ,

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

Injectivity of $V$ would imply $X Y = Y X$ for every pair of logical matrices, a contradiction.

Displayed formulas in this passage

unlabeled · line 1627
$π_{1} (X) V = V X, π_{2} (X) V = V X$
unlabeled · line 1635
$V X Y = π_{1} (X) π_{2} (Y) V = π_{2} (Y) π_{1} (X) V = V Y X .$

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section: The precise obstruction to duplicate logical algebras