From 1490514227d5884d6d9e6944772be7640466d100 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Wed, 26 Aug 2026 19:25:58 -0400 Subject: [PATCH] First draft of type decomposition. --- preamble.sty | 7 ++ src/op/vn/index.tex | 3 +- src/op/vn/projection.tex | 175 ++++++++++++++++++++++++++++---------- src/op/vn/type-decomp.tex | 174 +++++++++++++++++++++++++++++++++++++ src/op/vn/vn.tex | 6 ++ 5 files changed, 319 insertions(+), 46 deletions(-) create mode 100644 src/op/vn/type-decomp.tex diff --git a/preamble.sty b/preamble.sty index 4b5d653..c701d3a 100644 --- a/preamble.sty +++ b/preamble.sty @@ -232,3 +232,10 @@ \newcommand{\sotlim}{\operatorname*{\text{\small SOT}\text{-}\!\lim}} \newcommand{\wotlim}{\operatorname*{\text{\small WOT}\text{-}\!\lim}} +% VNA Types +\newcommand{\vnI}{\mathrm{I}} +\newcommand{\vnII}{\mathrm{II}} +\newcommand{\vnIIo}{\mathrm{II}_{1}} +\newcommand{\vnIIi}{\mathrm{II}_{\infty}} +\newcommand{\vnIII}{\mathrm{III}} + diff --git a/src/op/vn/index.tex b/src/op/vn/index.tex index 9f0eea9..4bc95d8 100644 --- a/src/op/vn/index.tex +++ b/src/op/vn/index.tex @@ -7,4 +7,5 @@ \input{./commutative.tex} \input{./spec.tex} \input{./fc.tex} -\input{./projection.tex} \ No newline at end of file +\input{./projection.tex} +\input{./type-decomp.tex} \ No newline at end of file diff --git a/src/op/vn/projection.tex b/src/op/vn/projection.tex index c862619..7868584 100644 --- a/src/op/vn/projection.tex +++ b/src/op/vn/projection.tex @@ -18,11 +18,22 @@ (2): For each $T \in A'$ and $Q \in S$, $TQ = QT$, so $Q(H)$ is a reducing subspace for $T$. As this holds for all $Q \in S$, $\bigcap_{Q \in S}Q(H)$ is a reducing subspace for $T$. Therefore $PT = TP$, and $P \in A$ by the \hyperref[Bicommutant Theorem]{theorem:bicommutant}. \end{proof} +\subsection{Central Support of Projections} +\label{subsection:projection-central-support} + + + \begin{definition}[Central Support] \label{definition:central-support-vna} Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P \in \text{Proj}(A)$, then $Z(P) = \inf_{Q \in \text{Proj}(Z(A)), Q \ge P}Q$ is the \textbf{central support} of $P$. \end{definition} +\begin{definition}[Centrally Orthogonal] +\label{definition:centrally-orthogonal} + Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $\seqi{P} \subset \text{Proj}(A)$, then $\seqi{P}$ is \textbf{centrally orthogonal} if $\bracsn{Z(P_i)}_{i \in I}$ is mutually orthogonal. +\end{definition} + + \begin{proposition} \label{proposition:central-support-vna} Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P \in \text{Proj}(A)$. For each $T \in A$, let $R(TP)$ be the orthogonal projection onto $\ol{TP(H)}$, then @@ -37,6 +48,65 @@ \end{proof} +\begin{proposition} +\label{proposition:central-support-mvn} + Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$, then the following are equivalent: + \begin{enumerate} + \item $Z(P)Z(Q) \ne 0$. + \item $PAQ \ne \bracsn{0}$. + \item There exists non-zero projections $P_0 \le P$ and $Q_0 \le Q$ such that $P_0 \sim Q_0$. + \end{enumerate} +\end{proposition} +\begin{proof}[Proof, {{\cite[Proposition 24.7]{Zhu}}}. ] + (1) $\Rightarrow$ (2): For each $T \in A$, let $R(T)$ be the orthogonal projection onto $\ol{T(H)}$. By \autoref{proposition:central-support-vna}, + \[ + Z(P) = \sup_{T \in A}R(TP) \quad Z(Q) = \sup_{T \in A}R(TQ) + \] + + so $Z(P)Z(Q) = \sup_{S, T \in A}R(SP)R(TQ) \ne 0$. Thus there exists $S, T \in A$ such that $R(SP)R(TQ) \ne 0$. As such, there exists $x, y \in H$ with + \[ + 0 \ne \dpn{SPx, TQy}{H} = \dpn{QT^*SPx, y}{H} + \] + + so $PAQ \ne 0$. + + (2) $\Rightarrow$ (3): Let $T \in A$ with $PTQ \ne 0$. Let $P_0 = R(PTQ)$ and $Q_0 = R(QT^*P)$, then $0 \ne P_0 \le P$, $0 \ne Q_0 \le Q$, and $P_0 \sim Q_0$ by \autoref{lemma:mvn-equivalent-adjoint}. + + (3) $\Rightarrow$ (1): By \autoref{lemma:central-support-mvn-eq}, $Z(P_0) = Z(Q_0)$, so + \[ + Z(P)Z(Q) = Z(P) \wedge Z(Q) \ge Z(P_0) \vee Z(Q_0) \ne 0 + \] +\end{proof} + +\begin{lemma} +\label{lemma:central-support-mvn-eq} + Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$ with $P \sim Q$, then: + \begin{enumerate} + \item $Z(P) = Z(Q)$. + \item For any central projection $R \in \text{Proj}(A)$, $PR \sim QR$. + \end{enumerate} +\end{lemma} +\begin{proof} + Let $V \in A$ with $P = V^*V$ and $Q = VV^*$, then $V$ is a partial isometry with initial space $P(H)$ and final space $Q(H)$. + + (1): Since $Z(P) \ge P$ and $Z(P) \in Z(A)$, + \[ + Z(P)Q = Z(P)VV^* = VZ(P)V^* = VV^* = Q + \] + + and $Z(P) \ge Q$, and $Z(P) \ge Z(Q)$. By symmetry, $Z(Q) \ge Z(P)$, so $Z(P) = Z(Q)$. + + (2): Let $R$ be a central projection, then + \[ + PR = V^*VR = V^*RV = (VR)^*(VR) \sim (VR)(VR)^* = VRV^* = VV^*R = QR + \] + + +\end{proof} + +\subsection{Murray-von Neumann Equivalence} +\label{subsection:mvn-equivalence} + \begin{lemma} \label{lemma:projection-mental-gymnastics} @@ -77,6 +147,29 @@ Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$, then $P$ is \textbf{Murrey-von Neumann subequivalent} to $Q$, denoted $P \preceq Q$, if there exists $R \in \text{Proj}(A)$ such that $P \sim R$ and $R \le Q$. \end{definition} +\begin{lemma} +\label{lemma:mvn-equivalent-direct-sum} + Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $\seqi{P}, \seqi{Q} \subset \text{Proj}(A)$ such that: + \begin{enumerate}[label=(\alph*)] + \item $\seqi{P}$ is mutually orthogonal. + \item $\seqi{Q}$ is mutually orthogonal. + \item For each $i \in I$, $P_i \sim Q_i$. + \end{enumerate} + + then $\sum_{i \in I}P_i \sim \sum_{i \in I}Q_i$. +\end{lemma} +\begin{proof} + For each $i \in I$, let $V_i \in A$ such that $P_i = V_i^*V_i$ and $Q_i = V_iV_i^*$, then $V_i$ is a partial isometry with initial space $P_i(H)$ and final space $Q_i(H)$. As $\seqi{P}$ is mutually orthogonal and $\seqi{Q}$ is mutually orthogonal, the sum $\sum_{i \in I}V_i$ converges in strong operator topology to an operator $V$, where + \[ + V^*V = \sum_{i, j \in I}V_i^*V_j = \sum_{i \in I}V_i^*V_i = \sum_{i \in I}P_i + \] + + and + \[ + VV^* = \sum_{i, j \in I}V_iV_j^* = \sum_{i \in I}V_iV_i^* = \sum_{i \in I}Q_i + \] +\end{proof} + \begin{lemma} \label{lemma:mvn-equivalent-adjoint} Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, $T \in A$, and $P, Q \in \text{Proj}(A)$ be orthogonal projections onto $\ol{T(H)}$ and $\ol{T^*(H)}$, respectively, then $P \sim Q$. @@ -132,50 +225,6 @@ by \autoref{lemma:mvn-equivalent-adjoint}. \end{proof} -\begin{lemma} -\label{lemma:central-support-mvn-eq} - Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$ with $P \sim Q$, then $Z(P) = Z(Q)$. -\end{lemma} -\begin{proof} - Let $V \in A$ with $P = V^*V$ and $Q = VV^*$, then $V$ is a partial isometry with initial space $P(H)$ and final space $Q(H)$. In which case, since $Z(P) \ge P$ and $Z(P) \in Z(A)$, - \[ - Z(P)Q = Z(P)VV^* = VZ(P)V^* = VV^* = Q - \] - - and $Z(P) \ge Q$, and $Z(P) \ge Z(Q)$. By symmetry, $Z(Q) \ge Z(P)$, so $Z(P) = Z(Q)$. -\end{proof} - - -\begin{proposition} -\label{proposition:central-support-mvn} - Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$, then the following are equivalent: - \begin{enumerate} - \item $Z(P)Z(Q) \ne 0$. - \item $PAQ \ne \bracsn{0}$. - \item There exists non-zero projections $P_0 \le P$ and $Q_0 \le Q$ such that $P_0 \sim Q_0$. - \end{enumerate} -\end{proposition} -\begin{proof}[Proof, {{\cite[Proposition 24.7]{Zhu}}}. ] - (1) $\Rightarrow$ (2): For each $T \in A$, let $R(T)$ be the orthogonal projection onto $\ol{T(H)}$. By \autoref{proposition:central-support-vna}, - \[ - Z(P) = \sup_{T \in A}R(TP) \quad Z(Q) = \sup_{T \in A}R(TQ) - \] - - so $Z(P)Z(Q) = \sup_{S, T \in A}R(SP)R(TQ) \ne 0$. Thus there exists $S, T \in A$ such that $R(SP)R(TQ) \ne 0$. As such, there exists $x, y \in H$ with - \[ - 0 \ne \dpn{SPx, TQy}{H} = \dpn{QT^*SPx, y}{H} - \] - - so $PAQ \ne 0$. - - (2) $\Rightarrow$ (3): Let $T \in A$ with $PTQ \ne 0$. Let $P_0 = R(PTQ)$ and $Q_0 = R(QT^*P)$, then $0 \ne P_0 \le P$, $0 \ne Q_0 \le Q$, and $P_0 \sim Q_0$ by \autoref{lemma:mvn-equivalent-adjoint}. - - (3) $\Rightarrow$ (1): By \autoref{lemma:central-support-mvn-eq}, $Z(P_0) = Z(Q_0)$, so - \[ - Z(P)Z(Q) = Z(P) \wedge Z(Q) \ge Z(P_0) \vee Z(Q_0) \ne 0 - \] -\end{proof} - \begin{theorem}["Cantor-Bernstein"] \label{theorem:murray-von-neumann-subequivalent-partial-order} @@ -205,7 +254,43 @@ &\sim Q_\infty + \sum_{n = 0}^\infty (Q_{2n + 1} - Q_{2n+2}) + \sum_{n = 0}^\infty (Q_{2n} - Q_{2n+1}) = Q \end{align*} - because $\sim$ is preserved through direct sums. + by \autoref{lemma:mvn-equivalent-direct-sum}. +\end{proof} + +\begin{theorem}[The Comparability Theorem] +\label{theorem:vna-comparability} + Let $H$ be a complex Hilbert space, $A \subset H$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$, then there exists a central projection $R$ such that $RP \preceq RQ$ and $(1 - R)Q \preceq (1 - R)P$. +\end{theorem} +\begin{proof}[Proof, {{\cite[Theorem 25.4]{Zhu}}}. ] + By Zorn's lemma, there exists maximal families $\seqi{P}, \seqi{Q} \subset \text{Proj}(A)$ such that: + \begin{enumerate}[label=(\roman*)] + \item $\seqi{P}$ is mutually orthogonal. + \item $\seqi{Q}$ is mutually orthogonal. + \item For each $i \in I$, $P_i \sim Q_i$. + \item For each $i \in I$, $P_i \le P$ and $Q_i \le Q$. + \end{enumerate} + + Let $P_0 = \sum_{i \in I}P_i$ and $Q_0 = \sum_{i \in I}Q_i$, then $P_0 \sim Q_0$ by \autoref{lemma:mvn-equivalent-direct-sum}. By maximality, there exists no non-zero $P', Q' \in \text{Proj}(A)$ such that $P' \le P - P_0$, $Q' \le Q - Q_0$, and $P' \sim Q'$. By \autoref{proposition:central-support-mvn}, $Z(P - P_0) Z(Q - Q_0) = 0$. + + Let $R = Z(Q - Q_0)$, then $Q - Q_0 \le R$ and $P - P_0 \le (I - R)$, so $(P - P_0)R = 0$ and $(Q - Q_0)R = Q - Q_0$. By \autoref{lemma:central-support-mvn-eq}, + \[ + PR = P_0R \sim Q_0R \le QR + \] + + and + \[ + Q(I - R) = Q_0(I - R) \sim P_0(I - R) \le P(I - R) + \] + +\end{proof} + +\begin{corollary} +\label{corollary:vna-factor-comparability} + Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a factor, then for any $P, Q \in \text{Proj}(A)$, either $P \prec Q$, $P \sim Q$, or $Q \prec P$. +\end{corollary} +\begin{proof} + By the \hyperref[comparability theorem]{theorem:vna-comparability}, there exists a central projection $R$ such that $PR \preceq QR$ and $Q(I - R) \preceq P(I - R)$. As $A$ is a factor, either $R = 0$ or $R = I$. In which case, $P \preqeq Q$ or $Q \preqeq P$. \end{proof} + diff --git a/src/op/vn/type-decomp.tex b/src/op/vn/type-decomp.tex new file mode 100644 index 0000000..9a2c268 --- /dev/null +++ b/src/op/vn/type-decomp.tex @@ -0,0 +1,174 @@ +\section{Type Decomposition} +\label{section:vna-type-decomposition} + + +\begin{definition}[Finite Projection] +\label{definition:finite-projection} + Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P \in \text{Proj}(A)$, then $P$ is \textbf{finite} if for any $Q \in \text{Proj}(A)$ with $P \sim Q$ and $Q \le P$, $P = Q$. For any $P \in \text{Proj}(A)$, $P$ is \textbf{infinite} if it is not finite. +\end{definition} + +\begin{definition}[Abelian Projection] +\label{definition:abelian-projection} + Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P \in \text{Proj}(A)$, then $P$ is \textbf{minimal} if $PAP$ is abelian. +\end{definition} + +\begin{definition}[Minimal Projection] +\label{definition:minimal-projection} + Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P \in \text{Proj}(A)$, then the following are equivalent: + \begin{enumerate} + \item $PAP = \complex P$. + \item There exists no $Q \in \text{Proj}(A)$ with $0 < Q < P$. + \end{enumerate} + + If the above holds, then $P$ is \textbf{minimal}. +\end{definition} +\begin{proof} + (1) $\Rightarrow$ (2): Let $Q \in \text{Proj}(A)$ with $Q \le P$, then $Q = PQP$. As $PAP = \complex P$, either $PQP = 0$ or $PQP = P$. + + (2) $\Rightarrow$ (1): Given that there exists no projections strictly between $0$ and $P$, the only projection in $PAP$ is $P$ itself. By \autoref{theorem:vn-projection-norm-dense}, the linear span of projections in $PAP$ is norm-dense in $PAP$. Therefore $PAP = \complex P$. +\end{proof} + +\begin{lemma} +\label{lemma:projection-types-gymnastics} + Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$. + \begin{enumerate} + \item If $P$ is minimal, then $P$ is abelian. + \item If $P$ is abelian, then $P$ is finite. + \item If $P$ is finite and $P \sim Q$, then $Q$ is finite. + \item If $P$ is finite and $Q \le P$, then $Q$ is finite. + \end{enumerate} +\end{lemma} +\begin{proof}[Proof, {{\cite[Section 26.1]{Zhu}}}. ] + (1): $PAP = \complex P$ is abelian. + + (2): Let $R \in \text{Proj}(A)$ with $P \sim R \le P$, then there exists $V \in A$ such that $R = V^*V$ and $P = VV^*$. Since $V$ has initial space $R(H) \subset P(H)$ and final space $P(H)$, $V = PVP$ and $V^* = PV^*P$. As $PAP$ is abelian, + \[ + R = V^*V = PV^*PPVP = PVPPV^*P = VV^* = P + \] + + (3): Let $R \in \text{Proj}(A)$ with $Q \sim R \le Q$. Let $V \in A$ with $Q = V^*V$ and $P = VV^*$, then $V$ is a partial isometry with initial space $Q(H)$ and final space $P(H)$. In which case, $P = VQV^*$, and $VRV^* \le VQV^* = P$. Let $U = (VRV^*)V$, then + \begin{align*} + U^*U &= (VRV^*V)^*(VRV^*V) = V^*VRV^* \cdot VRV^*V \\ + &= V^*VRV^*V = QRQ = R + \end{align*} + + + and as $Q = V^*PV$, + \begin{align*} + UU^* &= (VRV^*V)(VRV^*V)^* = VRV^*V \cdot V^*VRV^* \\ + &= VRV^*PVRV^* = VRQRV^* = VRV^* + \end{align*} + + so $VRV^* \sim R \sim Q \sim P$. Given that $P$ is finite, $VRV^* = P$. Therefore + \[ + R = QRQ = V^*VRV^*V = V^*PV = Q + \] + + (4): Let $R \in \text{Proj}(A)$ with $Q \sim R \le Q \le P$, then $P \sim (P - Q) + R \le P$, so $P - Q + R = P$, and $Q = R$. +\end{proof} + +\begin{lemma} +\label{lemma:centrally-orthogonal-sum-properties} + Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, $\seqi{P} \subset \text{Proj}(A)$ be centrally orthogonal, and $P = \sum_{i \in I}P_i$, then + \begin{enumerate} + \item For each $T \in A$, $PTP = \sum_{i \in I}P_iTP_i$. + \item If $\seqi{P}$ are abelian, then $P$ is also abelian. + \item If $\seqi{P}$ are finite, then $P$ is also finite. + \end{enumerate} +\end{lemma} +\begin{proof}[Proof, {{\cite[Lemma 26.2]{Zhu}}}. ] + (1): For each $i \in I$, $Z(P_i) \ge P_i$, so $Z(P_i)P_i = P_i$. For any $i, j \in I$ with $i \ne j$, $Z(P_i)$ and $Z(P_j)$ are orthogonal, so $Z(P_i)P_j = Z(P_i)Z(P_j)P_j = 0$. + + Let $T \in A$, then by \autoref{proposition:central-support-vna}, $Z(P_i)(H) \supset TP_i(H)$ for all $i \in I$. Therefore + \begin{align*} + PTP &= \sum_{i, j \in I}P_iTP_j = \sum_{i, j \in I}Z(P_i)P_i \cdot T \cdot Z(P_j)P_j \\ + &= \sum_{i, j \in I}Z(P_i)P_i \cdot Z(P_j) \cdot T \cdot Z(P_j)P_j \\ + &= \sum_{i \in I}Z(P_i)P_i \cdot T \cdot Z(P_i)P_i = \sum_{i \in I}P_i TP_i + \end{align*} + + (2): Let $S, T \in A$, then by (1), + \begin{align*} + PSP \cdot PTP &= \sum_{i, j \in I}P_iSP_i \cdot P_jSP_j = \sum_{i \in I}P_iSP_i \cdot P_iTP_i \\ + &= \sum_{i \in I}P_iTP_i \cdot P_iSP_i = PTP \cdot PSP + \end{align*} + + (3): Let $R \in \text{Proj}(A)$ with $P \sim R \le P$, and $V \in A$ with $R = V^*V$ and $P = VV^*$, then for each $i \in I$, $Z(P_i)R \sim Z(P_i)P = P_i$, and $Z(P_i)R \le Z(P_i)P = P_i$. As $\seqi{P}$ are finite, $Z(P_i)R = P_i$ for all $i \in I$. Therefore + \[ + R = RP = R\sum_{i \in I}Z(P_i)P_i = \sum_{i \in I}Z(P_i)RP_i = \sum_{i \in I}P_i = P + \] +\end{proof} + +\begin{definition}[Type $\vnI$] +\label{definition:vna-t1} + Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a von Neumann algebra, then $A$ is of \textbf{type $\vnI$} if for every non-zero central projection $P \in \text{Proj}(Z(A))$, there exists a non-zero abelian projection $Q \in \text{Proj}(A)$ with $P \ge Q$. +\end{definition} +% Todo: add a few equivalent characterisations. + +\begin{definition}[Type $\vnII$] +\label{definition:vna-t2} + Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a von Neumann algebra, then $A$ is of \textbf{type $\vnII$} if: + \begin{enumerate} + \item $A$ has no non-zero abelian projections. + \item For every non-zero central projection $P \in \text{Proj}(Z(A))$, there exists a non-zero finite projection $Q \in \text{Proj}(A)$ with $P \ge Q$. + \end{enumerate} +\end{definition} + +\begin{definition}[Type $\vnII_1$] +\label{definition:vna-t21} + Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a type $\vnII$ von Neumann algebra, then $A$ is of \textbf{type $\vnII_1$} if $I$ is a finite projection. +\end{definition} + +\begin{definition}[Type $\vnII_\infty$] +\label{definition:vna-t2inf} + Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a type $\vnII$ von Neumann algebra, then $A$ is of \textbf{type $\vnII_\infty$} if $A$ has no non-zero finite central projections. +\end{definition} + +\begin{definition}[Type $\vnIII$] +\label{definition:vna-t3} + Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a von Neumann algebra, then $A$ is of \textbf{type $\vnIII$} if $A$ has no non-zero finite projections. +\end{definition} + +\begin{theorem}[Type Decomposition] +\label{theorem:vna-type-decomposition} + Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a von Neumann algebra, then there exists unique von Neumann algebras $A_{\vnI}, A_{\vnII_1}, A_{\vnII_{\infty}}, A_{\vnIII} \subset A$\footnote{Not all four types are guaranteed to be present.} of type $\vnI$, $\vnII_1$, $\vnII_\infty$, and $\vnIII$, respectively, such that + \[ + A = A_{\vnI} \oplus A_{\vnII_1} \oplus A_{\vnII_{\infty}} \oplus A_{\vnIII} + \] +\end{theorem} +\begin{proof}[Proof, {{\cite[Theorem 26.3]{Zhu}}}. ] + ($\vnI$): By Zorn's lemma, there exists a maximal family $\seqi{P} \subset \text{Proj}(A)$ of centrally orthogonal abelian projections. Let $P = \sum_{i \in I}P_i$, then $P$ is abelian by (2) of \autoref{lemma:centrally-orthogonal-sum-properties}. + + Let $P_{\vnI} = Z(P)$, then $A_{\vnI} := P_{\vnI}AP_{\vnI}$ is a von Neumann algebra with identity $P_{\vnI}$. Let $R \in \text{Proj}(Z(A_{\vnI})) \setminus \bracs{0}$, then since $0 < R \le P_{\vnI}$, $RP \le R$ is a non-zero abelian projection. Therefore $A_{\vnI}$ is of type $\vnI$. + + ($\vnII$): Assume without loss of generality that $I \in A$. Since $\seqi{P}$ is maximal and $(I - P_{\vnI}) \in Z(A)$, $(I - P_{\vnI})A(I - P_{\vnI})$ has no non-zero abelian projections. + + By Zorn's lemma, there exists a maximal family $\seqj{Q} \subset \text{Proj}((I - P_{\vnI})A(I - P_{\vnI}))$ of centrally orthogonal finite projections. Let $Q = \sum_{j \in J}Q_j$, then $Q$ is finite by (3) of \autoref{lemma:centrally-orthogonal-sum-properties}. + + Let $P_{\vnII} = Z(Q)$ and $A_{\vnII} = P_{\vnII}AP_{\vnII}$, then $A_{\vnII}$ is a von Neumann algebra with identity $P_{\vnII}$. Let $R \in \text{Proj}(Z(A_{\vnII})) \setminus \bracs{0}$, then since $0 < R \le P_{\vnII}$, $RQ \le R$ is a non-zero finite projection by (4) of \autoref{lemma:projection-types-gymnastics}. + + ($\vnIII$): Let $P_{\vnIII} = I - P_{\vnI} - P_{\vnII}$ and $A_{\vnIII} = P_{\vnIII}AP_{\vnIII}$. Since $P_{\vnIII} \in Z(A)$ and $\seqi{P}$, $\seqi{Q}$ are maximal, $A_{\vnIII}$ has no non-zero finite projections. Therefore $A_{\vnIII}$ is of type $\vnIII$, and $A = A_{\vnI} \oplus A_{\vnII} \oplus A_{\vnIII}$. + + ($\vnII_1$): By Zorn's lemma, there exists a maximal family $\bracsn{R_k}_{k \in K} \subset \text{Proj}(A_{\vnII})$ of orthogonal central finite projections. Let $P_{\vnII_1} = \sum_{k \in K}R_k$, then $P_{\vnII_1}$ is a central finite projection by (3) of \autoref{lemma:centrally-orthogonal-sum-properties}. Hence $A_{\vnII_1} = P_{\vnII_1}AP_{\vnII_1}$ is of type $\vnII_1$. + + ($\vnII_\infty$): Let $P_{\vnII_\infty} = P_{\vnII} - P_{\vnII_1}$ and $A_{\vnII_\infty} = P_{\vnII_\infty}AP_{\vnII_\infty}$, then $P_{\vnII_\infty}$ is a central projection. By maximality of $\bracsn{R_k}_{k \in K}$, $A_{\vnII_\infty}$ admits no non-zero finite central projections. Therefore $A_{\vnII_\infty}$ is of type $\vnII_\infty$, $A_{\vnII} = A_{\vnII_1} \oplus A_{\vnII_\infty}$, and + \[ + A = A_{\vnI} \oplus A_{\vnII_1} \oplus A_{\vnII_{\infty}} \oplus A_{\vnIII} + \] + + (Uniqueness): Let $A = A_{\vnI}' \oplus A_{\vnII_1}' \oplus A_{\vnII_{\infty}}' \oplus A_{\vnIII}'$ be a decomposition of $A$ into von Neumann algebras of type $\vnI$, $\vnII_1$, $\vnII_\infty$, and $\vnIII$, respectively. + + Let $P_{\vnI}'$, $P_{\vnII_1}'$, $P_{\vnII_\infty}'$, and $P_{\vnIII_\infty}'$ be the identity elements of $A_{\vnI}$, $A_{\vnII_1}$, $A_{\vnII_{\infty}}$, and $A_{\vnIII}$, respectively, then + \[ + I = P_{\vnI}' \oplus P_{\vnII_1}' \oplus P_{\vnII_\infty}' \oplus P_{\vnIII_\infty}' + \] + + is an orthogonal direct sum, and + \begin{enumerate} + \item[($\vnI$)] Let $P_1 = P'_{\vnI}(I - P_{\vnI})$, then by construction of $P_{\vnI}$, there exists no non-zero abelian projection $R \in \text{Proj}(A)$ with $R \le P_1$. As both $P_{\vnI}'$ and $(I - P_{\vnI})$ are central, $P_1 \in A_{\vnI}'$, so $P_1 = 0$ because $A_{\vnI}'$ is of type $\vnI$. Thus $P_{\vnI}' \ge P_{\vnI}$. By symmetry, $P_{\vnI} = P_{\vnI}'$ and $A_{\vnI} = A_{\vnI}'$. + \item[($\vnII$, $\vnIII$)] Let $P'_{\vnII} = P_{\vnII_1} \oplus P_{\vnII_\infty}$ and $P_2 = P'_{\vnII}(I - P_{\vnI} - P_{\vnII})$. By construction of $P_{\vnII}$, there exists no non-zero finite projection $R \in \text{Proj}(A)$ with $R \le P_2$. Since $P_2 \in A_{\vnII}'$ and $A_{\vnII}'$ is of type $\vnII$, $P_2 = 0$ and $P_{\vnII}' \ge P_{\vnII}$. By symmetry, $P_{\vnII} = P_{\vnII}'$. Thus $P_{\vnIII} = P_{\vnIII}'$, $A_{\vnII} = A_{\vnII}'$, and $A_{\vnIII} = A_{\vnIII}'$. + \item[($\vnII_1$, $\vnII_\infty$)] Let $Q_2 = P'_{\vnII_1}(P_{\vnII} - P_{\vnII_1})$, then there exists no non-zero finite central projection $R \in \text{Proj}(A)$ with $R \le Q_2$. However, since $P'_{\vnII_1}$ is itself a central projection, every subprojection of $P'_{\vnII_1}$ is finite by (4) of \autoref{lemma:projection-types-gymnastics}, so $Q_2 = 0$, and $P_{\vnII_1}' \ge P_{\vnII_1}$. By symmetry, $P_{\vnII_1}' = P_{\vnII_1}$. Therefore $P_{\vnII_\infty} = P_{\vnII_\infty}$, $A_{\vnII_1}' = A_{\vnII_1}$, and $A_{\vnII_\infty}' = A_{\vnII_\infty}$. + \end{enumerate} + +\end{proof} + + diff --git a/src/op/vn/vn.tex b/src/op/vn/vn.tex index 4074efc..7841e1c 100644 --- a/src/op/vn/vn.tex +++ b/src/op/vn/vn.tex @@ -181,6 +181,12 @@ Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a $C^*$-subalgebra, then $A$ is a \textbf{von Neumann algebra acting on $H$} if $A$ is closed in the strong operator topology. \end{definition} +\begin{definition}[Factor] +\label{definition:vna-factor} + Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a von Neumann algebra, then $A$ is a \textbf{factor} if $Z(A) = \complex I$. +\end{definition} + + \begin{theorem}[Kaplansky Density Theorem] \label{theorem:kaplansky-density} Let $H$ be a Hilbert space, $A \subset B(H)$ be a $C^*$-subalgebra, and $B$ be the strong-operator closure of $A$, then: