Original text: CC0 1.0. Prerequisite proofs and component terms.

Bounded factorization

CC0 1.0.

Let $M\subseteq B(H)$ be a faithfully represented von Neumann algebra. In particular, one may take $M=B(H)$ for the resolvent-order proof. The lemma uses the bounded operator and Hilbert-space foundations.

Positivity refers to the inherited cone above. It does not assert that $\ell$ is bounded for the operator norm, or that it has already been extended to a normal weight on all of $M_+$. The order on these forms is positivity of their difference on this cone.

OA-MOD-DW-02 — A factorization inside the algebra

Lemma. If $x,y\in M$ and $y^*y\leq x^*x$, there is a unique contraction $v\in M$ which vanishes on $\overline{xH}^{\perp}$ and satisfies $y=vx$, in any faithful concrete realization $M\subseteq B(H)$.

Proof. On $xH$ define $x\xi\mapsto y\xi$. The inequality says both that this is well defined and that its norm is at most one. Extend it continuously to $\overline{xH}$, and set it equal to zero on the orthogonal complement. This proves existence and uniqueness in $B(H)$. Every unitary $u\in M'$ preserves $\overline{xH}$; because $x,y$ commute with $u$, both $v$ and $uvu^*$ have the prescribed properties. Thus $v$ commutes with every unitary of $M'$. Every element of a unital C*-algebra is a linear combination of unitaries: a self-adjoint contraction $b$ is the real part of $b+i(1-b^2)^{1/2}$. Consequently $v\in M''=M$. This uses bounded continuous functional calculus and the bicommutant theorem, with their stated hypotheses. $\square$

Editable source · Proof dependencies and component terms