From 56d081628f875e0a25d817d09d85b45f0a5741a8 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sun, 9 Aug 2026 16:08:57 -0400 Subject: [PATCH] Updated citation on the existence of projections. --- src/op/vn/vn.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/op/vn/vn.tex b/src/op/vn/vn.tex index bf730fa..2379745 100644 --- a/src/op/vn/vn.tex +++ b/src/op/vn/vn.tex @@ -23,7 +23,7 @@ \end{theorem} -\begin{proof}[Proof, {{\cite[Theorem 17.1]{Zhu}}}. ] +\begin{proof}[Proof, {{\cite[Section 17]{Zhu}}}. ] (1): After rescaling, assume without loss of generality that $-I \le T \le I$ for all $T \in \cf$. Since $\cf \subset A_{sa}$, $\norm{T}_{B(H)} = [T]_{sp} \le 1$ by \autoref{theorem:c-star-normal-spectral-radius}, where the spectral radius is taken with respect to $B(H)$. Thus $\cf \subset \ol{B_{A}(0, 1)}$, and is relatively compact in the weak operator topology by the \hyperref[Banach-Alaoglu Theorem]{proposition:bh-ultraweak-bounded}. As such, $\bigcap_{T \in \cf}\ol{\bracs{S \in \cf|S \ge T}}^{\text{\small WOT}} \ne \emptyset$. Let $R \in \bigcap_{T \in \cf}\ol{\bracs{S \in \cf|S \ge T}}^{\text{\small WOT}}$. Since $A$ is strong-operator closed, so is $A_{sa}$ by \autoref{proposition:bh-operator-topologies-facts}. Thus for each $T \in A_{sa}$, $\bracs{S \in A_{sa}|S \ge T}$ is closed in the weak operator topology, and $R \in A_{sa}$ with $R \ge T$ for all $T \in \cf$.