From cb4dcbaa188bef202ab9455fbabd2186c7a110a6 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Fri, 6 Nov 2026 15:52:47 -0500 Subject: [PATCH] ADP related house keeping. --- src/op/c-star/gns.tex | 2 ++ src/op/system/index.tex | 1 + src/op/vn/fc.tex | 35 +++++++++++++++++++++++++++++++++++ src/op/vn/index.tex | 1 + src/op/vn/type-decomp.tex | 6 ++++++ src/op/vn/vn.tex | 2 +- 6 files changed, 46 insertions(+), 1 deletion(-) diff --git a/src/op/c-star/gns.tex b/src/op/c-star/gns.tex index 46c44a0..b788c00 100644 --- a/src/op/c-star/gns.tex +++ b/src/op/c-star/gns.tex @@ -70,6 +70,8 @@ \end{enumerate} The representation $(H_\phi, \pi_\phi)$ is the \textbf{cyclic representation of $A$ induced by $\phi$}, and the triple $(H_\phi, \pi_\phi, \xi_\phi)$ is the \textbf{Gelfand-Naimark-Segal (GNS) triple associated with $\phi$}. + + If $\phi$ is faithful, then $L^2(M, \phi)$ also denotes $H_\phi$. \end{definition} \begin{proof}[Proof, {{\cite[Proposition 14.2]{Zhu}}}. ] (2): Fix $x \in A$, then for each $y_1, y_2 \in A$ with $y_1 - y_2 \in N_\phi$, $x(y_1 - y_2) \in N_\phi$ by \autoref{lemma:cstar-state-kernel}, so $\pi_\phi^0(x)$ is well-defined on $A/N_\phi$. diff --git a/src/op/system/index.tex b/src/op/system/index.tex index 582d25a..94daef7 100644 --- a/src/op/system/index.tex +++ b/src/op/system/index.tex @@ -12,4 +12,5 @@ \input{./os-mn.tex} \input{./arveson.tex} \input{./aos.tex} +\input{./tensor.tex} diff --git a/src/op/vn/fc.tex b/src/op/vn/fc.tex index 5097331..0e3ba6b 100644 --- a/src/op/vn/fc.tex +++ b/src/op/vn/fc.tex @@ -41,6 +41,10 @@ Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a von Neumann algebra, then: \begin{enumerate} \item For each normal operator $T \in A$ and $f \in B^\infty(\sigma_{B(H)}(T); \complex)$ with $f(0) = 0$, $f(T) \in A$. + \item For each $T \in A$ with $0 \le T \le 1$, there exists projections $\seq{P_n} \subset A$ such that + \[ + T = \sum_{n = 1}^\infty \frac{P_n}{2^n} + \] \item The linear span of projections in $A$ is norm dense in $A$. \end{enumerate} \end{theorem} @@ -52,6 +56,20 @@ If $B$ is arbitrary, then $\one_{B \setminus B_\complex(0, r)} \to \one_B$ in the weak* topology of $C(\sigma_{B(H)}(T); \complex)^{**}$ as $r \downto 0$. As $\one_{B \setminus B_{\complex}(0, r)}(T) \in A$ for all $r > 0$, $\one_B(T) \in A$ as well. By linearity, $g(T) \in A$ for all $g \in \Sigma(\sigma_{B(H)}(T); \complex)$ with $g(0) = 0$. By \autoref{lemma:separable-metric-space-approx-identity}, $\bracsn{g \in \Sigma(\sigma_{B(H)}(T); \complex)|g(0) = 0}$ is uniformly dense in $\bracsn{f \in B^\infty(\sigma_{B(H)(T)}; \complex)|f(0) = 0}$. Therefore $f(T) \in A$ for all $f \in B^\infty(\sigma_{B(H)}(T); \complex)$ with $f(0) = 0$. + + + + (2): Let $f = \text{Id}: [0, 1] \to [0, 1]$, then + \[ + f = \sum_{n = 1}^\infty \frac{\one_{\bracsn{x \in [0, 1]| \lceil 2^nx \rceil = 0 \mod 2}}}{2^n} + \] + + Let $E: \cb_{\sigma_{A}(T)} \to \text{Proj}(A)$ be the resolution of the identity for $f$, then by the \hyperref[$L^\infty$ functional calculus]{definition:linfty-functional-calculus}, + \begin{align*} + T &= f(T) = \sum_{n = 1}^\infty\frac{\one_{\bracsn{x \in [0, 1]| \lceil 2^nx \rceil = 0 \mod 2}}(T)}{2^n} \\ + &= \sum_{n = 1}^\infty 2^{-n}E(\bracsn{x \in [0, 1]| \lceil 2^nx \rceil = 0 \mod 2}) + \end{align*} + \end{proof} \begin{theorem} @@ -77,4 +95,21 @@ (2): In particular, as $e^{t\phi}(\partial B(0, 1)) \subset \partial B(0, 1)$, the spectrum of $e^{t\phi}$ as an element in the domain of the $L^\infty$ functional calculus, is contained in $\partial B(0, 1)$. Thus if $T$ is unitary, then the path $t \mapsto e^{t\phi(T)}$ lies in the unitary group of $A$ by \autoref{corollary:spectrum-characterisation-iff}. \end{proof} +\begin{proposition} +\label{proposition:vna-two-sided-ideal} + Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $I \subset M$ be a two-sided ideal, then: + \begin{enumerate} + \item $I$ is self-adjoint. + \item Let $x \in I$ with $x > 0$, $E: \cb_{[0, \infty)} \to \text{Proj}(M)$ be its resolution of the identity, and $t > 0$, then $E([t, \infty)) \in I$. + \item If $I$ is weak-operator closed, then there exists a unique projection $z \in Z(M)$ with $I = Mz$. + \end{enumerate} +\end{proposition} +\begin{proof}[Proof, {{\cite[Proposition 2.4.15]{AnantharamanPopa}}}. ] + (1): Let $x = u|x|$ be the \hyperref[polar decomposition]{theorem:hilbert-polar-decomposition} of $x$, then $|x| = u^*x \in I$, and $x^* = |x|u^* \in I$. + + (2): Let $f: [0, \infty) \to \real$ be defined by $f(s) = s^{-1}\one_{\bracsn{s \ge t}}(s)$, then by the \hyperref[Borel functional calculus]{definition:linfty-functional-calculus}, $E([t, \infty)) = xf(x) \in I$. + + (3): By (5) of \autoref{theorem:existence-of-projections-vna}. +\end{proof} + diff --git a/src/op/vn/index.tex b/src/op/vn/index.tex index 4bc95d8..b3aba3f 100644 --- a/src/op/vn/index.tex +++ b/src/op/vn/index.tex @@ -4,6 +4,7 @@ \input{./topologies.tex} \input{./cayley.tex} \input{./vn.tex} +\input{./weak.tex} \input{./commutative.tex} \input{./spec.tex} \input{./fc.tex} diff --git a/src/op/vn/type-decomp.tex b/src/op/vn/type-decomp.tex index 49240a1..3e001b6 100644 --- a/src/op/vn/type-decomp.tex +++ b/src/op/vn/type-decomp.tex @@ -28,6 +28,12 @@ (2) $\Rightarrow$ (1): Given that there exist no projections strictly between $0$ and $P$, the only non-zero 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{definition}[Diffuse] +\label{definition:diffuse-vna} + Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, then $M$ is \textbf{diffuse} if it admits no minimal projections. +\end{definition} + + \begin{lemma} \label{lemma:projection-types-gymnastics} Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $P, Q \in \text{Proj}(A)$. diff --git a/src/op/vn/vn.tex b/src/op/vn/vn.tex index e24a5db..da512e5 100644 --- a/src/op/vn/vn.tex +++ b/src/op/vn/vn.tex @@ -5,7 +5,7 @@ \label{theorem:existence-of-projections-vna} Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a strong-operator closed $C^*$-subalgebra, then: \begin{enumerate} - \item For any bounded directed family $\cf \subset A_{sa}$, $\sup(\cf) = \sotlim_{T \in \cf}T \in A_{sa}$. + \item For any bounded, upward-directed family $\cf \subset A_{sa}$, $\sup(\cf) = \sotlim_{T \in \cf}T \in A_{sa}$. \item For any directed family of projections $\mathcal{P} \subset A_{sa}$, $\sup(\mathcal{P}) \in A$ is the projection onto $\ol{\bigcup_{P \in \mathcal{P}}P(H)}$. \end{enumerate}