Corners and inherited normal functionals

Retained original OA-MOD-CP-11 proof under its exact published CC0 component grant. Scoped selection and bindings by GPT-6.1 Sol (OpenAI), Ultra, October 2026. The original manuscript names its OA-MOD-CP authorship and supplies no individual model attribution in this excerpt.

This is the complete corner-predual statement used by the normal-products lesson. The published trace-class/predual chapter, Theorems 6.2 and 9.4, supplies vector-series descriptions, relative ultraweak topology, completeness and isometric duality. The companion spatial tensor supplement, Proposition 3.1, gives an alternative vector-series construction. This proof applies to every projection, including zero and noncentral projections. The statement concerns inherited normal functionals on a corner.

OA-MOD-CP-11 — Corners have exactly their inherited predual

Let $p\in M$ be a projection and let $N=pMp$ act on $pH$. It is a concrete von Neumann algebra there. To verify closedness, extend a weak operator convergent net on $pH$ by zero on $(1-p)H$. The extensions converge weakly on $H$, so a net from $pMp$ has its limit in $M$ and still satisfies $x=pxp$.

Let $j:N\to M$ be this inclusion, and let $C:M\to N$ be $C(x)=pxp$. Both are contractions and $Cj=\operatorname{id}_N$. A vector-series functional on $N$ extends through $C$ using the same sequences in $H$. Restricting a series from $M$ through $j$ replaces both vector sequences by their images under $p$. Hence the ultraweak topology on $N$ is exactly the inherited topology.

The maps $$r:M_*\to N_*,\quad r(f)=f\circ j,\qquad s:N_*\to M_*,\quad s(g)=g\circ C$$ are contractions, $rs=1$, and $s$ is isometric: restriction gives the reverse norm inequality. They preserve positivity. Every $g\in N_*$ has an extension of exactly its norm, and therefore $$N_*\cong M_*/\ker r \cong\operatorname{ran}P_*, \qquad P_*f(x)=f(pxp), \tag{CP.17}$$ isometrically, where $P_*=sr$ is a contractive idempotent. The second identification uses $s$; a quotient is not identified with a subspace without a specified map.

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