Manuscript En03-A@1 · En03-A.tex · lemma · 5398614e

Lemma: One-layer support

Kinematics and causal cuts on a prescribed schedule › Scheduled blocks · lines 656–673 · proof included or linked

En03-A@1#lem:one-layer, En03-A@1#eq:one-layer-support

If $Φ = ⨂_{B \in B} Φ_{B}$ and $A \in A (Y)$ , then

supp Φ^{*} (A) \subseteq ⋃ {B \in B : B \cap Y \neq = \emptyset} .

Proof. Let $Y^{'}$ be the displayed union. Its complement is a union of blocks. Unitality on those blocks gives

Φ^{*} (A_{Y^{'}} \otimes I_{X ∖ Y^{'}}) = Φ_{Y^{'}}^{*} (A_{Y^{'}}) \otimes I_{X ∖ Y^{'}} .

Displayed formulas in this passage

En03-A@1#eq:one-layer-support · line 659
$supp Φ^{*} (A) \subseteq ⋃ {B \in B : B \cap Y \neq = \emptyset} .$
unlabeled · line 669
$Φ^{*} (A_{Y^{'}} \otimes I_{X ∖ Y^{'}}) = Φ_{Y^{'}}^{*} (A_{Y^{'}}) \otimes I_{X ∖ Y^{'}} .$