Theorem: Injective partial-contraction presentations

The maps

are linear isomorphisms onto their actual-functional images.

Proof. If $q_L(v)=0$, Proposition prop:relations gives ${\partial }_t(vK)=0$. The distribution $vK$ is compactly supported in $M_{+}^2$. Lemma lem:primitive gives $vK=0$.

If the smooth function $v$ were nonzero at a point, it would have a smooth reciprocal on a nonempty neighborhood. There $K=v^{-1}(vK)=0$, contradicting Lemma lem:fullsupport. Thus $v=0$, and the algebraic tensor embedding proves injectivity. Surjectivity onto the named image is its definition.

For the right map, use (eq:rightrelation); conjugation and exchange of variables preserve full support.