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Bokuan Li
2026-08-20 16:25:48 -04:00
parent b3d3620ec9
commit 881f4a4746
6 changed files with 71 additions and 29 deletions
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4
src/op/c-star/gns.tex
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@@ -127,7 +127,7 @@
is a representation of $A$, which is injective if for every $x \in A$, there exists $\phi \in \mathcal{S}$ with $\dpn{x^*x, \phi}{A} \ne 0$.
In particular, $A$ is isomorphic to a closed subalgebra of $B([l^2(P(A)); H_\phi])$.
In particular, $A$ is isomorphic to a closed subalgebra of $B([l^2(PS(A)); H_\phi])$.
\end{enumerate}
\end{theorem}
\begin{proof}
@@ -168,7 +168,7 @@
so $\pi_\phi(x) \ne 0$, and $\pi_{\mathcal{S}}(x) \ne 0$ as well.
By \autoref{corollary:cstar-positive-weakstar-dense}, for each $x \in A$, there exists $\phi \in P(A)$ with $\dpn{x^*x, \phi}{A} \ne 0$, so $\pi_{P(A)}$ is injective. By \autoref{theorem:continuity-of-homomorphism-c-star}, $\pi_{P(A)}(A)$ is closed in $B([l^2(P(A)); H_\phi])$.
By \autoref{corollary:cstar-positive-weakstar-dense}, for each $x \in A$, there exists $\phi \in PS(A)$ with $\dpn{x^*x, \phi}{A} \ne 0$, so $\pi_{PS(A)}$ is injective. By \autoref{theorem:continuity-of-homomorphism-c-star}, $\pi_{PS(A)}(A)$ is closed in $B([l^2(PS(A)); H_\phi])$.
\end{proof}

44
src/op/c-star/state.tex
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@@ -31,12 +31,12 @@
\begin{definition}[Pure State]
\label{definition:pure-state}
Let $A$ be a unital $C^*$-algebra and $\phi \in S(A)$, then $\phi$ is a \textbf{pure state} if $\phi$ is an extreme point of $S(A)$. The set $P(A)$ is the collection of all pure states of $A$.
Let $A$ be a unital $C^*$-algebra and $\phi \in S(A)$, then $\phi$ is a \textbf{pure state} if $\phi$ is an extreme point of $S(A)$. The set $PS(A)$ is the collection of all pure states of $A$.
\end{definition}
\begin{proposition}
\label{proposition:state-space-compact-convex}
Let $A$ be a unital $C^*$-algebra, then $S(A)$ is a compact convex set, and $S(A)$ is the weak*-closed convex hull of $P(A)$.
Let $A$ be a unital $C^*$-algebra, then $S(A)$ is a compact convex set, and $S(A)$ is the weak*-closed convex hull of $PS(A)$.
\end{proposition}
\begin{proof}
Since the evaluation map is weak* continuous and
@@ -48,15 +48,15 @@
By \autoref{theorem:cstar-positive-algebraic}, $S(A) \subset \ol{B_{A^*}(0, 1)}$, which is weak* compact by the \hyperref[Banach-Alaoglu Theorem]{theorem:alaoglu}. Therefore $S(A)$ is compact by \autoref{proposition:compact-extensions}.
By the \hyperref[Krein-Milman Theorem]{theorem:krein-milman}, $S(A)$ is the weak*-closed convex hull of $P(A)$.
By the \hyperref[Krein-Milman Theorem]{theorem:krein-milman}, $S(A)$ is the weak*-closed convex hull of $PS(A)$.
\end{proof}
\begin{proposition}
\label{proposition:multiplicative-pure-state}
Let $A$ be a unital $C^*$-algebra, then:
\begin{enumerate}
\item $\Omega(A) \subset P(A)$.
\item If $A$ is commutative, then $\Omega(A) = P(A)$.
\item $\Omega(A) \subset PS(A)$.
\item If $A$ is commutative, then $\Omega(A) = PS(A)$.
\end{enumerate}
\end{proposition}
\begin{proof}
@@ -80,7 +80,7 @@
Let $A$ be a unital $C^*$-algebra, $B \subset A$ be a $C^*$-subalgebra with $1_A \in B$, and $\phi \in S(B)$, then
\begin{enumerate}
\item There exists $\Phi \in S(A)$ such that $\Phi|_B = \phi$.
\item If $\phi \in P(B)$, then there exists $\Phi \in P(A)$ such that $\Phi|_B = \phi$.
\item If $\phi \in PS(B)$, then there exists $\Phi \in PS(A)$ such that $\Phi|_B = \phi$.
\end{enumerate}
\end{theorem}
\begin{proof}
@@ -88,7 +88,7 @@
(2): Let $E(\phi) = \bracs{\Phi \in S(A)|\Phi|_B = \phi}$ be the collection of all extensions of $\phi$, then $E(\phi)$ is a weak*-closed convex subset of $S(A)$. By (1), $E(\phi)$ is non-empty, and as such admits an extreme point $\Phi$ by the \hyperref[Krein-Milman Theorem]{theorem:krein-milman}.
Let $\psi, \rho \in S(A)$ and $t \in (0, 1)$ such that $\Phi = (1 - t)\psi + t\rho$. In which case, $\phi = (1 - t)\psi|_B + t\rho|_B$. Since $\phi \in P(B)$, $\phi = \psi|_B = \rho|_B$, so $\psi, \rho \in E(\phi)$. As $\Phi$ is an extreme point of $E(\phi)$, $\Phi = \psi = \rho$. Therefore $\Phi \in P(A)$.
Let $\psi, \rho \in S(A)$ and $t \in (0, 1)$ such that $\Phi = (1 - t)\psi + t\rho$. In which case, $\phi = (1 - t)\psi|_B + t\rho|_B$. Since $\phi \in PS(B)$, $\phi = \psi|_B = \rho|_B$, so $\psi, \rho \in E(\phi)$. As $\Phi$ is an extreme point of $E(\phi)$, $\Phi = \psi = \rho$. Therefore $\Phi \in PS(A)$.
\end{proof}
@@ -96,15 +96,15 @@
\label{corollary:cstar-positive-property-probe}
Let $A$ be a unital $C^*$-algebra and $x \in A$ be normal, then\footnote{The crude bound seems kind of tragic, but it wouldn't be true otherwise. }
\begin{align*}
\sigma_A(x) &\subset \bracs{\dpn{x, \phi}{A}|\phi \in P(A)} \\
\sigma_A(x) &\subset \bracs{\dpn{x, \phi}{A}|\phi \in PS(A)} \\
&\subset \bracs{\dpn{x, \phi}{A}|\phi \in S(A)} = \ol{\text{Conv}}(\sigma_A(x))
\end{align*}
In particular, there exists $\phi \in P(A)$ such that $\norm{x}_A = |\dpn{x, \phi}{A}|$.
In particular, there exists $\phi \in PS(A)$ such that $\norm{x}_A = |\dpn{x, \phi}{A}|$.
\end{corollary}
\begin{proof}
Let $\lambda \in \sigma_A(x)$. By \autoref{proposition:gelfand-transform-gymnastics}, there exists $\phi \in \Omega(A[x])$ such that $\dpn{x, \phi}{A[x]} = \lambda$. By \autoref{proposition:multiplicative-pure-state}, $\phi \in P(A[x])$. The \hyperref[pure state extension theorem]{theorem:cstar-pure-state-extension} implies that there exists $\Phi \in P(A)$ such that $\Phi|_{A[x]} = \phi$. Thus $\Phi$ is a pure state with $\dpn{x, \Phi}{A} = \lambda$, and $ \sigma_A(x) \subset \bracs{\dpn{x, \Phi}{A}|\Phi \in P(A)}$.
Let $\lambda \in \sigma_A(x)$. By \autoref{proposition:gelfand-transform-gymnastics}, there exists $\phi \in \Omega(A[x])$ such that $\dpn{x, \phi}{A[x]} = \lambda$. By \autoref{proposition:multiplicative-pure-state}, $\phi \in PS(A[x])$. The \hyperref[pure state extension theorem]{theorem:cstar-pure-state-extension} implies that there exists $\Phi \in PS(A)$ such that $\Phi|_{A[x]} = \phi$. Thus $\Phi$ is a pure state with $\dpn{x, \Phi}{A} = \lambda$, and $ \sigma_A(x) \subset \bracs{\dpn{x, \Phi}{A}|\Phi \in PS(A)}$.
Let $\Phi \in S(A)$ and $\phi = \Phi|_{A[x]}$, then $\phi \in S(A[x])$ as well. By the \hyperref[Gelfand-Naimark Theorem]{theorem:gelfand-naimark}, the \hyperref[Spectral Theorem]{theorem:spectral-c-star}, and the \hyperref[Riesz Representation Theorem]{theorem:riesz-radon}, $\phi$ takes the form of a Radon probability measure $\mu$ on $\sigma_A(x)$. In which case,
\[
@@ -113,7 +113,7 @@
Finally, since $S(A)$ is compact and convex by \autoref{proposition:state-space-compact-convex},
\begin{align*}
\bracs{\dpn{x, \phi}{A}|\phi \in S(A)} &= \ol{\text{Conv}}(\bracs{\dpn{x, \phi}{A}|\phi \in P(A)}) \\
\bracs{\dpn{x, \phi}{A}|\phi \in S(A)} &= \ol{\text{Conv}}(\bracs{\dpn{x, \phi}{A}|\phi \in PS(A)}) \\
&\subset \ol{\text{Conv}}(\sigma_A(x))
\end{align*}
@@ -139,35 +139,35 @@
\label{corollary:cstar-positive-weakstar-dense}
Let $A$ be a unital $C^*$-algebra, then:
\begin{enumerate}
\item For each $x \in A$, $x = 0$ if and only if $\dpn{x, \phi}{A} = 0$ for all $\phi \in P(A)$.
\item The linear span of $P(A)$ is weak*-dense in $A^*$.
\item For each $x \in A$, $x = 0$ if and only if $\dpn{x, \phi}{A} = 0$ for all $\phi \in PS(A)$.
\item The linear span of $PS(A)$ is weak*-dense in $A^*$.
\end{enumerate}
Moreover, for any $x \in A$,
\begin{enumerate}[start=2]
\item $x$ is self-adjoint if and only if $\dpn{x, \phi}{A} \in \real$ for all $\phi \in P(A)$.
\item $x$ is positive if and only if $\dpn{x, \phi}{A} \ge 0$ for all $\phi \in P(A)$.
\item $x$ is self-adjoint if and only if $\dpn{x, \phi}{A} \in \real$ for all $\phi \in PS(A)$.
\item $x$ is positive if and only if $\dpn{x, \phi}{A} \ge 0$ for all $\phi \in PS(A)$.
\end{enumerate}
\end{corollary}
\begin{proof}[Proof, {{\cite[Theorem 13.9]{Zhu}}}. ]
(1): Let $x \in A$ such that $\dpn{x, \phi}{A} = 0$ for all $\phi \in P(A)$. First suppose that $x$ is self-adjoint. By \autoref{theorem:cstar-state-existence}, $\sigma_A(x) = \bracs{0}$, and $\norm{x}_A = [x]_{sp} = 0$ by \autoref{theorem:c-star-normal-spectral-radius}.
(1): Let $x \in A$ such that $\dpn{x, \phi}{A} = 0$ for all $\phi \in PS(A)$. First suppose that $x$ is self-adjoint. By \autoref{theorem:cstar-state-existence}, $\sigma_A(x) = \bracs{0}$, and $\norm{x}_A = [x]_{sp} = 0$ by \autoref{theorem:c-star-normal-spectral-radius}.
Now suppose that $x$ is arbitrary. In this case, for each $\phi \in P(A)$,
Now suppose that $x$ is arbitrary. In this case, for each $\phi \in PS(A)$,
\[
0 = \text{Re}(\dpn{x, \phi}{A}) = \dpn{\text{Re}(x), \phi}{A}
\]
because $\phi$ is Hermitian. Similarly, $\dpn{\text{Im}(x), \phi}{A} = 0$ as well. Thus $\text{Re}(x) = \text{Im}(x) = 0$, and $x = 0$ as well.
(2): Since the linear span of $P(A)$ separates points in $A$, it is weak*-dense in $A^*$ by \autoref{lemma:duality-dense}.
(2): Since the linear span of $PS(A)$ separates points in $A$, it is weak*-dense in $A^*$ by \autoref{lemma:duality-dense}.
(3): Let $\phi \in P(A)$, then $\phi$ is Hermitian. If $x$ is self-adjoint, then $\dpn{x, \phi}{A} \in \real$.
(3): Let $\phi \in PS(A)$, then $\phi$ is Hermitian. If $x$ is self-adjoint, then $\dpn{x, \phi}{A} \in \real$.
On the other hand, if $\dpn{x, \phi}{A} \in \real$, then $\dpn{x, \phi}{A} = \dpn{x^*, \phi}{A}$, and $\dpn{x - x^*, \phi}{A} =0 $. If this holds for all $\phi \in P(A)$, then $x - x^* = 0$ by (1), and $x$ is self-adjoint.
On the other hand, if $\dpn{x, \phi}{A} \in \real$, then $\dpn{x, \phi}{A} = \dpn{x^*, \phi}{A}$, and $\dpn{x - x^*, \phi}{A} =0 $. If this holds for all $\phi \in PS(A)$, then $x - x^* = 0$ by (1), and $x$ is self-adjoint.
(4): Let $\phi \in P(A)$, then $\phi$ is positive. Thus if $x$ is positive, $\dpn{x, \phi}{A} \ge 0$.
(4): Let $\phi \in PS(A)$, then $\phi$ is positive. Thus if $x$ is positive, $\dpn{x, \phi}{A} \ge 0$.
On the other hand, if $\dpn{x, \phi}{A} \ge 0$ for all $\phi \in P(A)$, then $x$ is self-adjoint by (3). By \autoref{corollary:cstar-positive-property-probe}, $\sigma_A(x) \subset [0, \infty)$. As such, $x$ is positive by \autoref{corollary:spectrum-characterisation-iff}.
On the other hand, if $\dpn{x, \phi}{A} \ge 0$ for all $\phi \in PS(A)$, then $x$ is self-adjoint by (3). By \autoref{corollary:cstar-positive-property-probe}, $\sigma_A(x) \subset [0, \infty)$. As such, $x$ is positive by \autoref{corollary:spectrum-characterisation-iff}.
\end{proof}

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src/op/example/bounded.tex
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@@ -18,7 +18,7 @@
\begin{proposition}
\label{proposition:partial-isometry-characterisation}
Let $H$ be a complex Hilbert space and $T \in B(H)$, then $T$ is a partial isometry if and only if $T^*T$ is a projection.
Let $H$ be a complex Hilbert space and $T \in B(H)$, then $T$ is a partial isometry if and only if $T^*T$ is a projection. In which case, $T^*T$ is a projection onto $\ker(T)^\perp$.
\end{proposition}
\begin{proof}
($\Rightarrow$): Suppose that $T$ is a partial isometry. Let $x \in \ker(T)^\perp$, then $\dpn{Tx, Tx}{H} = \norm{x}_H^2$ and $\dpn{T^*Tx, x}{H} = \norm{x}_H^2$. By \hyperref[polarisation]{proposition:polarisation-complex}, for each $x, y \in \ker(T)^\perp$,
@@ -27,10 +27,18 @@
&= \frac{1}{4}\sum_{k = 0}^3 i^k \dpn{x + i^ky, x + i^ky}{H} = \dpn{x, y}{H}
\end{align*}
Therefore $T^*T$ is idempotent. As $T^*T$ is self-adjoint, it is a projection.
Therefore $T^*T$ is idempotent. As $T^*T$ is self-adjoint, it is a projection onto $\ker(T)^\perp$.
($\Leftarrow$): Suppose that $T^*T$ is a projection, then for each $x \in \ker(T)^\perp$, $\dpn{Tx, Tx}{H} = \dpn{T^*Tx, x}{H} = \norm{x}_H^2$.
\end{proof}
\begin{corollary}
\label{corollary:partial-isometry-adjoint}
Let $H$ be a complex Hilbert space and $T \in B(H)$, then $T$ is a partial isometry if and only if $T^*$ is a partial isometry.
\end{corollary}
\begin{proof}[Proof, {{\cite[Corollary 12.7]{Zhu}}}. ]
Suppose that $T$ is a partial isometry, then $P = T^*T$ is a projection onto $\ker(T)^\perp$ by \autoref{proposition:partial-isometry-characterisation}. In which case, $T(T^*T) = T$ and $(TT^*)^2 = T(T^*T)T^* = TT^*$, so $TT^*$ is a projection, and $T^*$ is a partial isometry by \autoref{proposition:partial-isometry-characterisation}.
\end{proof}
\begin{theorem}[Polar Decomposition]
\label{theorem:hilbert-polar-decomposition}

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src/op/notation.tex
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@@ -16,7 +16,7 @@
$\Gamma = \Gamma_A$ & The Gelfand transform on $A$. & \autoref{definition:gelfand-transform} \\
$A[S]$ & $C^*$-subalgebra of $A$ generated by $S \subset A$. & \autoref{definition:generated-subalgebra} \\
$S(A)$ & State space of a $C^*$-algebra $A$. & \autoref{definition:cstar-state} \\
$P(A)$ & Pure state space of a $C^*$-algebra $A$. & \autoref{definition:pure-state} \\
$PS(A)$ & Pure state space of a $C^*$-algebra $A$. & \autoref{definition:pure-state} \\
$\dpn{x, y}{\phi}$ & Defined as $\dpn{y^*x, \phi}{A}$, the pseudo inner product associated to a positive linear functional. & \autoref{definition:cstar-state-pseudo-inner-product} \\
$(H_\phi, \pi_\phi, \xi_\phi)$ & GNS triple associated with $\phi \in S(A)$. & \autoref{definition:gns-triple} \\
$U(T)$ & Cayley transform of $T$. & \autoref{definition:cayley-transform-bounded} \\

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src/op/vn/projection.tex Normal file
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@@ -0,0 +1,34 @@
\section{The Projection Lattice}
\label{section:vn-projection-lattice}
\begin{definition}[Projection Lattice]
\label{definition:vn-projection-lattice}
Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a von Neumann algebra, then the set of all projections $\text{Proj}(A)$ in $A$ is the \textbf{projection lattice} of $A$.
\end{definition}
\begin{proposition}
\label{proposition:vn-projection-lattice-complete}
Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a von Neumann algebra, then $\text{Proj}(A)$ is order complete.
\end{proposition}
\begin{proof}
By \autoref{theorem:existence-of-projections-vna}.
\end{proof}
\begin{definition}[Murray-von Neumann Equivalent]
\label{definition:murrey-von-neuman-equivalent}
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 There exists $V \in A$ such that $P = V^*V$ and $Q = VV^*$.
\item There exists a partial isometry $V \in A$ from $P(H)$ to $Q(H)$.
\end{enumerate}
If the above holds, then $P$ and $Q$ are \textbf{Murrey-von Neumann equivalent}, denoted $P \sim Q$. The relation $\sim$ is an equivalence relation on $\text{Proj}(A)$.
\end{definition}
\begin{proof}
(1) $\Rightarrow$ (2): Let $V \in A$ with $P = V^*V$ and $Q = VV^*$. By \autoref{proposition:partial-isometry-characterisation}, $V$ is a partial isometry with initial space $\ker(P)^\perp$, and $V^*$ is a partial isometry with initial space $\ker(Q)^\perp$. Therefore $V$ is a partial isometry from $P(H)$ to $Q(H)$.
(2) $\Rightarrow$ (1): By \autoref{proposition:partial-isometry-characterisation}, $P = V^*V$ is a projection onto $\ker(V)^\perp$, and $Q = VV^*$ is a projection onto $\ker(V^*)^\perp = V(H)$.
\end{proof}

4
src/op/vn/spec.tex
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@@ -292,7 +292,7 @@
\label{corollary:commutative-von-neumann-linfty}
Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a commutative von Neumann algebra with $I \in A$, then:
\begin{enumerate}
\item There exists a LCH space $\Omega$ and a decomposable Radon measure $\mu$ on $\Omega$ such that $A$ is *-isomorphic to $L^\infty(\Omega; \complex)$.
\item There exists a LCH space $\Omega$ and a decomposable Radon measure $\mu$ on $\Omega$ such that $A$ is *-isomorphic to $L^\infty(\mu; \complex)$.
\item If $A$ admits a cyclic vector, then $\Omega$ may be taken to be compact.
\item If $H$ is separable, then $\Omega$ may be taken to be compact.
\end{enumerate}
@@ -309,7 +309,7 @@
(2): By \hyperref[Spectral Theorem II]{theorem:spectral-theorem-vn-2}, there exists a single positive Radon measure $\mu \in \mathscr{E}$ on $\Omega(A)$ such that $\mathscr{E}$ is absolutely continuous with respect to it. Therefore the index set in (1) can be taken to be a singleton.
(3): If $H$ is separable, then so is $\mathscr{E}$. As such, there exists a single positive Radon measure on $\mu \in \mathscr{E}$ on $\Omega(A)$ such that $\mathscr{E}$ is absolutely continuous with respect to it. Therefore the index set in (1) can be taken to be a singleton.
(3): If $H$ is separable, then so is $\mathscr{E}$. As such, there exists a single positive Radon measure $\mu \in \mathscr{E}$ on $\Omega(A)$ such that $\mathscr{E}$ is absolutely continuous with respect to it. Therefore the index set in (1) can be taken to be a singleton.
\end{proof}