From e5ef0d51df583c263955bc46ee6fe5a97a0428f9 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Wed, 8 Jul 2026 15:02:23 -0400 Subject: [PATCH] Added states. --- src/fa/norm/hilbert.tex | 9 +- src/op/c-star/index.tex | 3 +- src/op/c-star/positive.tex | 9 ++ src/op/c-star/state.tex | 163 +++++++++++++++++++++++++++++++++++++ src/op/notation.tex | 3 +- 5 files changed, 182 insertions(+), 5 deletions(-) create mode 100644 src/op/c-star/state.tex diff --git a/src/fa/norm/hilbert.tex b/src/fa/norm/hilbert.tex index 75effbd..fd7109a 100644 --- a/src/fa/norm/hilbert.tex +++ b/src/fa/norm/hilbert.tex @@ -83,20 +83,23 @@ \begin{definition}[Inner Product] \label{definition:inner-product} - Let $E$ be a vector space over $K$ and $\inp_E: E \times E \to K$, then $\inp_E$ is an \textbf{inner product} if: + Let $E$ be a vector space over $K$ and $\inp_E: E \times E \to K$, then $\inp_E$ is a \textbf{pseudo inner product} if: \begin{enumerate}[label=(H\arabic*)] \item For each $x, y, z \in E$, $\angles{x + y, z}_E = \dpn{x, z}{E} + \dpn{y, z}{E}$. \item For any $x, y \in E$ and $\mu \in K$, $\dpn{\mu x, y}{E} = \mu \dpn{x, y}{E}$. \item For every $x, y \in E$, $\dpn{x, y}{E} = \ol{\dpn{y, x}{E}}$. - \item[(I)] For each $x \in E$, $\dpn{x, x}{E} \ge 0$, with equality if and only if $x = 0$. + \item[(I)] For each $x \in E$, $\dpn{x, x}{E} \ge 0$. \end{enumerate} + and an \textbf{inner product} if for each $x \in E$, $\dpn{x, x}{E} = 0$ if and only if $x = 0$. + + \end{definition} \begin{proposition}[Cauchy-Schwarz Inequality] \label{proposition:cauchy-schwarz} - Let $H$ be a vector space over $K \in \RC$ and $\inp_H: E \times E \to K$ be an inner product, then for any $x, y \in H$, $\dpn{x, y}{H} \le \norm{x}_H\norm{y}_H$. + Let $H$ be a vector space over $K \in \RC$ and $\inp_H: E \times E \to K$ be a pseudo inner product, then for any $x, y \in H$, $\dpn{x, y}{H} \le \norm{x}_H\norm{y}_H$. \end{proposition} \begin{proof}[Proof, {{\cite[Theorem 5.19]{Folland}}}. ] Assume without loss of generality that $\dpn{x, y}{H} > 0$, then for each $t \in \real$, diff --git a/src/op/c-star/index.tex b/src/op/c-star/index.tex index bcf7b55..a755692 100644 --- a/src/op/c-star/index.tex +++ b/src/op/c-star/index.tex @@ -9,4 +9,5 @@ \input{./gelfand.tex} \input{./cont.tex} \input{./order.tex} -\input{./positive.tex} \ No newline at end of file +\input{./positive.tex} +\input{./state.tex} \ No newline at end of file diff --git a/src/op/c-star/positive.tex b/src/op/c-star/positive.tex index 64f98ee..de6e4e9 100644 --- a/src/op/c-star/positive.tex +++ b/src/op/c-star/positive.tex @@ -29,4 +29,13 @@ \end{proof} +\begin{corollary} +\label{corollary:positive-linear-functional-extension} + Let $A$ be a unital $C^*$-algebra, $B \subset A$ be a closed subspace with $1_A \in B$, and $\phi \in B^*$ with $\norm{\phi}_{B^*} = \dpn{1_A, \phi}{B}$, then there exists a positive linear functional $\Phi \in A^*$ such that $\Phi|_B = A$. +\end{corollary} +\begin{proof} + By the \hyperref[Hahn-Banach Theorem]{theorem:hahn-banach}, there exists $\Phi \in A^*$ such that $\Phi|_B = A$ and $\norm{\Phi}_{A^*} = \norm{\phi}_{B^*}$. In which case, $\norm{\Phi}_{A^*} = \dpn{1_A, \Phi}{A}$, and $\Phi$ is also positive by \autoref{theorem:cstar-positive-algebraic}. +\end{proof} + + diff --git a/src/op/c-star/state.tex b/src/op/c-star/state.tex new file mode 100644 index 0000000..1ee3f98 --- /dev/null +++ b/src/op/c-star/state.tex @@ -0,0 +1,163 @@ +\section{States} +\label{section:cstar-states} + + +\begin{definition}[State] +\label{definition:cstar-state} + Let $A$ be a unital $C^*$-algebra and $\phi \in A^*$, then $\phi$ is a \textbf{state} if $\phi$ is positive and $\dpn{1, \phi}{A} = 1$. + + The set of states $S(A) \subset A^*$ of $A$ equipped with the weak* topology is the \textbf{state space} of $A$. +\end{definition} + +\begin{lemma} +\label{lemma:cstar-state-cauchy-schwarz} + Let $A$ be a unital $C^*$-algebra and $\phi \in A^*$ be a positive linear functional, then the mapping + \[ + A \times A \to \complex \quad (x, y) \mapsto \dpn{x, y}{\phi} := \dpn{y^*x, \phi}{A} + \] + + is a pseudo inner product. In particular, for any $x, y \in A$, + \[ + |\dpn{y^*x, \phi}{A}|^2 = |\dpn{x, y}{\phi}|^2 \le \dpn{x, x}{\phi} \cdot \dpn{y, y}{\phi} + \] +\end{lemma} +\begin{proof} + By the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}. +\end{proof} + + + +\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 $A$. +\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 $P(A)$. +\end{proposition} +\begin{proof} + Since the evaluation map is weak* continuous and + \[ + S(A) = \bracs{\phi \in A^*|\dpn{1, \phi}{A} = 1} \cap \bigcap_{\substack{x \in A \\ x \ge 0}}\bracs{\phi \in A^*|\dpn{x, \phi}{A} \ge 0} + \] + + the state space is an intersection of convex and weak*-closed sets, so it is closed and convex. + + 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 $P(A)$. +\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 $\Omega(A) = P(A)$. + \end{enumerate} +\end{proposition} +\begin{proof} + (1): Let $\phi \in \Omega(A)$. By \autoref{proposition:multiplicative-unit}, $\norm{\phi}_{A^*} = \dpn{1, \phi}{A} = 1$. Thus $\phi$ is a state by \autoref{theorem:cstar-positive-algebraic}, and $\Omega(A) \subset S(A)$. + + Let $\psi, \rho \in S(A)$ and $t \in (0, 1)$ such that $\phi = (1 - t)\psi + t\rho$, then for each $x \in \ker(\phi)$, $x^*x \in \ker(\phi)$ as well. As $t \ne 0$, $x^*x \in \ker(\psi)$ and $x^*x \in \ker(\rho)$. By the \hyperref[Cauchy-Schwarz inequality]{lemma:cstar-state-cauchy-schwarz}, + \[ + |\dpn{x, \psi}{A}|^2 = |\dpn{1^*x, \psi}{A}|^2 \le \dpn{1, \psi}{A} \cdot \dpn{x^*x, \psi}{A} = 0 + \] + + Likewise, $\dpn{x, \rho}{A} = 0$ as well. Hence $\ker(\psi), \ker(\rho) \supset \ker(\phi)$. Thus there exist scalars $\alpha, \beta \in \complex$ such that $\phi = \alpha \psi = \beta \rho$. However, since $\phi, \psi, \rho \in S(A)$, $\alpha = \beta = 1$, and $\phi = \psi = \rho$. Therefore $\phi$ is a pure state. + + (2): Using the \hyperref[Gelfand-Naimark Theorem]{theorem:gelfand-naimark}, identify $A$ with $C(\Omega(A); \complex)$ and $S(A)$ as Radon probability measures on $\Omega(A)$. + + Let $\cm = \bracs{t\mu|\mu \in S(A), t \in [0, 1]}$. By \autoref{proposition:space-of-measures-extreme-points}, the extreme points of $\cm$ are the delta masses $\bracs{\delta_x|x \in \Omega(A)}$, and possibly $0$. For any $\mu \in S(A)$, $\nu, \rho \in \cm$, and $t \in (0, 1)$, $\mu = (1 - t)\nu + t\rho$ implies that $\nu(\Omega(A)) = \rho(\Omega(A)) = 1$, and $\nu, \rho \in S(A)$ as well. Thus the extreme points of $S(A)$ are exactly the delta masses $\bracs{\delta_x|x \in \Omega(A)}$, which correspond to $\Omega(A)$ itself. +\end{proof} + + +\begin{theorem}[Extension of States] +\label{theorem:cstar-pure-state-extension} + 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 $\Phi|_B = \phi$. + \end{enumerate} +\end{theorem} +\begin{proof} + (1): By \autoref{theorem:cstar-positive-algebraic}, $\norm{\phi}_{B^*} = \dpn{1_A, \phi}{B}$. By the \hyperref[Hahn-Banach Theorem]{theorem:hahn-banach}, there exists $\Phi \in A^*$ such that $\Phi|_B = \phi$ and $\norm{\Phi}_{A^*} = \norm{\phi}_{B^*} = \dpn{1_A, \Phi}{A}$. Thus \autoref{theorem:cstar-positive-algebraic} implies that $\Phi \in S(A)$. + + (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 $\Phi \in P(A)$. +\end{proof} + + +\begin{theorem} +\label{theorem:cstar-state-existence} + Let $A$ be a unital $C^*$-algebra, $x \in A$, and $\lambda \in \sigma_A(x)$, then there exists $\phi \in S(A)$ such that $\dpn{x, \phi}{A} = \lambda$. +\end{theorem} +\begin{proof}[Proof, {{\cite[Theorem 13.7]{Zhu}}}. ] + Let $B = \text{span}\bracs{x, 1}$. For each $\alpha x + \beta \in B$, let $\dpn{\alpha x + \beta, \phi_0}{B} = \alpha \lambda + \beta$. Since $\sigma_A(1) = \bracs{1}$, $\phi_0 \in B^*$ is a well-defined linear functional with $\dpn{x, \phi_0}{B} = \lambda$ and $\dpn{1, \phi_0}{B} = 1$. + + In addition, for each $\alpha x + \beta \in B$, $\alpha \lambda + \beta \in \sigma_A(\alpha x + \beta)$ by \autoref{proposition:commutative-spectrum-gymnastics}, and + \[ + |\alpha \lambda + \beta| \le [\alpha x + \beta]_{sp} \le \norm{\alpha x + \beta}_A + \] + + Thus $\norm{\phi_0}_{B^*} = \dpn{1, \phi_0}{B} = 1$. By the \hyperref[Hahn-Banach Theorem]{theorem:hahn-banach}, there exists $\phi \in A^*$ such that $\phi|_B = \phi_0$ and $\norm{\phi}_{A^*} = \norm{\phi_0}_{B^*}$. In which case, $\dpn{x, \phi}{A} = \lambda$ and $\dpn{1, \phi}{A} = \norm{\phi}_{A^*} = 1$. By \autoref{theorem:cstar-positive-algebraic}, $\phi$ is positive and hence a state. +\end{proof} + +\begin{corollary} +\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 $A^*$. + \end{enumerate} +\end{corollary} +\begin{proof} + (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}. + + Now suppose that $x$ is arbitrary. In this case, for each $\phi \in P(A)$, + \[ + 0 = \text{Re}(\dpn{x, \phi}{A}) = \dpn{\text{Re}(x), \phi}{A} + \] + + 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}. +\end{proof} + +\begin{corollary} +\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)} \\ + &\subset \bracs{\dpn{x, \phi}{A}|\phi \in S(A)} = \ol{\text{Conv}}(\sigma_A(x)) + \end{align*} + + + In particular, + \begin{enumerate} + \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 There exists $\phi \in P(A)$ such that $\norm{x}_A = |\dpn{x, \phi}{A}|$. + \end{enumerate} +\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 $\Phi \in S(A)$ 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, + \[ + \dpn{x, \Phi}{A} = \dpn{x, \phi}{A[x]} = \int_{\sigma_A(x)}\lambda \mu(d\lambda) \in \ol{\text{Conv}}(\sigma_A(x)) + \] + + 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)}) \\ + &\subset \ol{\text{Conv}}(\sigma_A(x)) + \end{align*} + + by \autoref{proposition:compact-extensions} and \autoref{proposition:closure-of-image}. + + (1), (2): By \autoref{corollary:spectrum-characterisation-iff}. +\end{proof} + diff --git a/src/op/notation.tex b/src/op/notation.tex index 9e6914c..8fedf08 100644 --- a/src/op/notation.tex +++ b/src/op/notation.tex @@ -15,7 +15,8 @@ $\cm(A)$ & Maximal ideal space of $A$. & \autoref{definition:maximal-ideal} \\ $\Gamma = \Gamma_A$ & 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} \\ $M_n(\complex)$ & Algebra of $n \times n$ matrices over $\complex$. & \autoref{definition:matrix-algebra} \\ $B(H)$ & Algebra of bounded operators on a Hilbert space. & \autoref{definition:hilbert-endomorphism} \\ $A(D)$ & The disk algebra. & \autoref{definition:disk-algebra} \\