Manuscript XI-E@1 · XI-E.tex · proposition · 46e55514

Proposition: Change of product data

Route 1: membership, change of product data, and the loop stop › Independence up to isomorphism · lines 929–953 · proof included or linked

XI-E@1#prop:alpha

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The map

α_{d} = exp (\frac{ℏ}{2} ⟨ d, \frac{δ ^{2}}{δ Φ ^{2}} ⟩)

is an algebra isomorphism $E_{H} \to E_{H^{'}}$ , fixing the linear generators and having inverse $α_{- d}$ .

Proof. Smooth self-contractions against compact kernels exist and preserve the polynomial domain. The second-order operator has the graded product rule whose cross term is twice $Γ_{d}$ . Exponentiation gives

α_{d} (F ⋆_{H} G) = α_{d} F ⋆_{H^{'}} α_{d} G .

The bisolution property makes the map descend to the free shell. The two exponentials with opposite $d$ compose to the identity.

Displayed formulas in this passage

unlabeled · line 932
$α_{d} = exp (\frac{ℏ}{2} ⟨ d, \frac{δ ^{2}}{δ Φ ^{2}} ⟩)$
unlabeled · line 946
$α_{d} (F ⋆_{H} G) = α_{d} F ⋆_{H^{'}} α_{d} G .$

Proposition: Change of product data

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