First draft of B(H).
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Bokuan Li
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\item If $P$ is abelian, then $P$ is finite.
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\item If $P$ is finite and $P \sim Q$, then $Q$ is finite.
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\item If $P$ is finite and $Q \le P$, then $Q$ is finite.
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\item If $P$ is minimal and $P \sim Q$, then $Q$ is minimal.
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\item If $P, Q$ are minimal with $P \sim Q$, then for any $U, V \in A$ with $P = U^*U = V^*V$ and $Q = UU^* = U^*U$, there exists $\lambda \in \partial B_{\complex}(0, 1)$ such that $P = \lambda Q$.
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\end{enumerate}
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\end{lemma}
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\begin{proof}[Proof, {{\cite[Section 26.1]{Zhu}}}. ]
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@@ -65,6 +67,28 @@
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\]
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(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 $Q = R$.
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(5): Since $P \sim Q$, there exists $V \in A$ with $P = V^*V$ and $Q = VV^*$. Let $R \in \text{Proj}(A)$ with $0 \le R \le Q$, then $0 \le V^*RV \le V^*QV = P$. By minimality of $P$, $V^*RV = P$, so
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\[
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R = QRQ = VV^*RVV^* = VPV^* = Q
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\]
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(6): Let $R = U^*V$, then since $U$ and $V$ are partial isometries with initial space $P(H)$ and final space $Q(H)$,
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\[
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PRP = U^*U \cdot U^*V \cdot V^*V = U^*QV = U^*V
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\]
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so $PRP \in PAP = \complex P$. Thus there exists $\lambda \in \complex$ such that $R = \lambda P$. In which case,
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\[
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\lambda U = U \cdot \lambda P = UU^*V = QV = V
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\]
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and
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\[
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P = V^*V \sim \lambda \ol{\lambda} U^*U = |\lambda|^2 Q
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\]
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so $|\lambda| = 1$.
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\end{proof}
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\begin{lemma}
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@@ -171,4 +195,99 @@
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\end{proof}
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\begin{lemma}
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\label{lemma:abelian-minimal-factor}
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Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a factor, and $P \in \text{Proj}(A)$ be abelian, then $P$ is minimal.
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\end{lemma}
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\begin{proof}
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Let $Q \in \text{Proj}(A)$ with $0 \le Q \le P$, then by the \hyperref[comparability theorem]{corollary:vna-factor-comparability}, either $Q \preceq P - Q$ or $P - Q \preceq Q$. Assume without loss of generality that $Q \preceq P - Q$.
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Let $V \in A$ such that $Q = V^*V$ and $VV^* \le P - Q$, then $V$ is a partial isometry with initial and final spaces contained in $P(H)$. Since $P$ is abelian, $V \in PAP$, and $Q = V^*V = VV^* \le P - Q$. Therefore $Q = 0$, and $P$ is minimal.
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\end{proof}
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\begin{lemma}
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\label{lemma:type1-bh-matrix}
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Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a factor, $T \in A$, $\seqi{P} \subset \text{Proj}(A)$ be non-zero minimal projections such that $I = \sum_{i \in I}P_i$, and $\bracsn{V_{i, j}}_{i, j \in I} \subset A$ such that $P_i = V_{i, j}^*V_{i, j}$ and $P_j = V_{i, j}V_{i, j}^*$ for all $i, j \in I$, then there exists $\bracsn{\mu_{i, j}}_{i, j \in I} \subset \complex$ such that $T = \sum_{i, j \in I}\mu_{i, j}V_{i, j}$.
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\end{lemma}
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\begin{proof}
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Let $i, j \in I$, then since $P_i$ and $P_j$ are minimal,
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\[
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(P_iTP_j)^*(P_iTP_j) = P_jT^*TP_j \in \complex P_j \quad (P_iTP_j)(P_iTP_j)^* = P_iTT^*P_i \in \complex P_i
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\]
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If $P_iTP_j \ne 0$, then $P_jT^*TP_j$ and $P_iTT^*P_i$ are positive, and there exists $\lambda > 0$ such that $\lambda P_iTP_j$ is a partial isometry with initial space $P_j(H)$ and final space $P_i(H)$. By \autoref{lemma:projection-types-gymnastics}, there exists $\mu_{i, j} \in \complex$ such that $P_iTP_j = \mu V_{j, i}$. Therefore
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\[
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T = \sum_{i, j \in I}P_jTP_i = \sum_{i, j \in I}\mu_{i, j}V_{i, j}
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\]
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\end{proof}
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\begin{theorem}
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\label{theorem:type1-bh}
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Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a factor, then the following are equivalent:
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\begin{enumerate}
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\item $A$ is of type $\vnI$.
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\item There exists a minimal projection $P \in \text{Proj}(A)$.
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\item For every non-zero $P \in \text{Proj}(A)$, there exists a non-zero minimal projection $Q \in \text{Proj}(A)$ such that $P \ge Q$.
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\item There exists a complex Hilbert space $K$ and a *-isomorphism $\pi: A \to B(K)$.
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\end{enumerate}
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\end{theorem}
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\begin{proof}
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(1) $\Rightarrow$ (2): Since $A$ is of type $\vnI$, $A$ admits an abelian projection $Q \in \text{Proj}(A)$. By \autoref{lemma:abelian-minimal-factor}, $Q$ is minimal.
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(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 $Q \preceq P$.
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If $P \preceq Q$, then there exists $R \in \text{Proj}(A)$ with $P \sim R \le Q$. In which case, $R$ is minimal, and $P$ is also minimal by (5) of \autoref{lemma:projection-types-gymnastics}.
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If $Q \preceq P$, then there exists $R \in \text{Proj}(A)$ with $Q \sim R \le P$. By (5) of \autoref{lemma:projection-types-gymnastics}, $R$ is minimal with $R \le P$.
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(3) $\Rightarrow$ (4): By Zorn's lemma, there exists a maximal orthogonal family $\seqi{P} \subset \text{Proj}(A)$ of minimal projections. Since $\seqi{P}$ is maximal and every non-zero projection admits a non-zero minimal subprojection, $I = \sum_{i \in I}P_i$, and $H = \bigoplus_{i \in I}P_iH$.
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For each $i, j \in I$, by the \hyperref[comparability theorem]{corollary:vna-factor-comparability}, either $P_i \preceq P_j$ or $P_j \preceq P_i$. In both cases, since both projections are minimal, $P_i \sim P_j$. Thus there exists a partial isometry $V_{i, j} \in A$ with initial space $P_i(H)$ and final space $P_j(H)$ such that $P_i = V_{i, j}^*V_{i, j}$ and $P_j = V_{i, j}V_{i, j}^*$. By fixing a particular family\footnote{The partial isometries need not to be unique. }, assume without loss of generality that $V_{i, j}^* = V_{j, i}$ for all $i, j \in I$.
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Fix $i_0 \in I$, let $H_0 = P_{i_0}I$, and define
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\[
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U: H \to l^2(I; H_0) \quad (Ux)_i = V_{i, i_0}P_ix
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\]
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then since $H = \bigoplus_{i \in I}P_iH$ and each $V_{i, i_0}$ is a partial isometry, $U$ is an isometry with inverse
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\[
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U^{-1}: l^2(I; H_0) \to H \quad U^{-1}x = \sum_{i \in I}V_{i_0, i}x_i
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\]
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For each $i \in I$, denote $e_i = \one_{\bracs{i}} \in l^2(I; \complex)$, then for every $x \in l^2(I; H_0)$,
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\[
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UP_iU^{-1}x = U P_i\sum_{i \in I} V_{i_0, i}x_i = e_{i}V_{i, i_0}P_iV_{i_0, i}x_i = e_iP_{i_0}x_i = e_ix_i
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\]
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so $UP_iU^{-1}$ is the projection onto the $i$-th component of $l^2(I; H_0)$.
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Let $T \in A$, then by \autoref{lemma:type1-bh-matrix}, there exists $\bracsn{\mu_{i, j}}_{i, j \in I} \subset \complex$ such that $T = \sum_{i, j \in I}\mu_{i,j}V_{i, j}$. In which case, for any $x \in l^2(I; \complex)$ and $v \in H$,
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\[
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UTU^{-1}(xv) = UT \sum_{i \in I}V_{i_0, i}x_i v = U\sum_{i, j, k \in I}\mu_{i, j}x_i \cdot V_{i, j}V_{i_0, k} \cdot v
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\]
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For each $i, j \in I$, there exists $\lambda_{i, j} \in \partial B_\complex(0, 1)$ such that $V_{i, j}V_{i_0, i} = \lambda_{i, j}V_{i_0, j}$ by (6) of \autoref{lemma:projection-types-gymnastics}. For every $i \in I$, let $e_i = \one_{\bracs{i}} \in l^2(I; \complex)$, then
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\begin{align*}
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UTU^{-1}(xv) &=U\sum_{i, j, k \in I}\mu_{i, j}x_i \cdot V_{i, j}V_{i_0, k} \cdot v = U\sum_{i, j \in I}\mu_{i, j}x_i \cdot V_{i, j}V_{i_0, i} \cdot v \\
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&= U\sum_{i, j \in I}\lambda_{i, j}\mu_{i, j}x_i \cdot V_{i_0, j} \cdot v = \sum_{i, j \in I}\lambda_{i, j}\mu_{i, j}x_i \cdot e_j \cdot V_{j, i_0}P_jV_{i_0, j} \cdot v \\
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&= \sum_{i, j \in I}\lambda_{i, j}\mu_{i, j}x_i \cdot e_j \cdot v \in l^2(I; \complex v)
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\end{align*}
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Moreover, if $v \ne 0$, then $UTU^{-1}(xv) = 0$ for all $x \in l^2(I; \complex)$ implies that $\mu_{i, j} =0 $ for all $i, j \in I$, and $T = 0$. Thus for any $v \in H_0 \setminus \bracs{0}$, the mapping
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\[
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\pi: A \to B(l^2(I; \complex v)) \quad T \mapsto UTU^{-1}|_{l^2(I; \complex v)}
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\]
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is an injective $*$-homomorphism.
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Finally, since $\ol{B_A(0, 1)}$ is weak-operator compact and $U$ is an isometry, $\pi(\ol{B_A(0, 1)})$ is also weak-operator compact, and $\pi(A) \subset B(l^2(I; \complex v))$ is a von Neumann algebra. Let $i, j \in I$, then for each $x \in l^2(I; \complex)$,
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\[
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\pi(V_{i, j})(xv) = x_i \cdot e_j \cdot V_{j, i_0}V_{i, j}V_{i_0, i} \cdot v = \lambda_{i, j}x_i \cdot e_j \cdot V_{j, i_0}V_{i_0, i} = \lambda_{i, j}x_i \cdot e_j \cdot v
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\]
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so $\pi(V_{i, j}) = \lambda_{i, j} e_jv \otimes e_iv$. As $\bracsn{e_iv}_{i \in I}$ is an orthonormal basis for $l^2(I; \complex v)$, $\pi(A) = B(l^2(I; \complex v))$.
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\end{proof}
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