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Added citation for the type 1 proof.
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Bokuan Li
2026-08-27 20:30:50 -04:00
parent f340ab9a29
commit f165b4fd2d
2 changed files with 11 additions and 1 deletions
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10
refs.bib
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@@ -283,3 +283,13 @@
volume = {2}, volume = {2},
year = {1951} year = {1951}
} }
@MISC {TownesType1,
TITLE = {Classification of Type 1 factors},
AUTHOR = {Leslie Townes (https://math.stackexchange.com/users/18076/leslie-townes)},
HOWPUBLISHED = {Mathematics Stack Exchange},
NOTE = {URL:https://math.stackexchange.com/q/150258 (version: 2012-05-27)},
EPRINT = {https://math.stackexchange.com/q/150258},
URL = {https://math.stackexchange.com/q/150258}
}

2
src/op/vn/type-decomp.tex
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@@ -234,7 +234,7 @@
\item There exists a complex Hilbert space $K$ and a *-isomorphism $\pi: A \to B(K)$. \item There exists a complex Hilbert space $K$ and a *-isomorphism $\pi: A \to B(K)$.
\end{enumerate} \end{enumerate}
\end{theorem} \end{theorem}
\begin{proof} \begin{proof}[Proof, {{\cite{TownesType1}}}. ]
(1) $\Rightarrow$ (2): Since $A$ is of type $\vnI$, $A$ admits a non-zero abelian projection $Q \in \text{Proj}(A)$. By \autoref{lemma:abelian-minimal-factor}, $Q$ is minimal. (1) $\Rightarrow$ (2): Since $A$ is of type $\vnI$, $A$ admits a non-zero abelian projection $Q \in \text{Proj}(A)$. By \autoref{lemma:abelian-minimal-factor}, $Q$ is minimal.
(2) $\Rightarrow$ (3): Let $P \in \text{Proj}(A)$ and $Q \in \text{Proj}(A)$ be a minimal projection. By the \hyperref[comparability theorem]{corollary:vna-factor-comparability}, either $P \preceq Q$ or $Q \preceq P$. (2) $\Rightarrow$ (3): Let $P \in \text{Proj}(A)$ and $Q \in \text{Proj}(A)$ be a minimal projection. By the \hyperref[comparability theorem]{corollary:vna-factor-comparability}, either $P \preceq Q$ or $Q \preceq P$.