From de9b6fb813a8ed0a1c5d9b68508e2e2a4c93c2b5 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Tue, 7 Jul 2026 17:26:52 -0400 Subject: [PATCH] Added characterisation of positive linear functionals. --- src/fa/tvs/complexify.tex | 25 +++++++++++++++++++++++++ src/op/banach/spectrum.tex | 4 ++-- src/op/c-star/index.tex | 3 ++- src/op/c-star/positive.tex | 31 +++++++++++++++++++++++++++++++ src/op/c-star/sa.tex | 4 ++-- 5 files changed, 62 insertions(+), 5 deletions(-) create mode 100644 src/op/c-star/positive.tex diff --git a/src/fa/tvs/complexify.tex b/src/fa/tvs/complexify.tex index 6881b47..351eef5 100644 --- a/src/fa/tvs/complexify.tex +++ b/src/fa/tvs/complexify.tex @@ -195,3 +195,28 @@ \end{proof} + + +\begin{proposition} +\label{proposition:hermitian-functional-norm} + Let $E$ be a normed vector space over $\complex$, $*: E \to E$ be a complex conjugation map such that $\norm{x}_E = \normn{x^*}_E$ for all $x \in E$, and $\phi \in E^*$ be a Hermitian functional, then + \[ + \norm{\phi}_{E^*} = \sup\bracsn{\dpn{x, \phi}{E}|x \in E, x = x^*, \norm{x}_E = 1} + \] +\end{proposition} +\begin{proof} + Since $\bracsn{x \in E|x = x^*} \subset E$, $\norm{\phi}_{E^*} \ge \sup\bracsn{\dpn{x, \phi}{E}|x \in E, x = x^*, \norm{x}_E = 1}$. + + On the other hand, let $x \in E$ with $\norm{x}_E = 1$. Assume without loss of generality that $\dpn{x, \phi}{E} \in \real$, then + \begin{align*} + \dpn{x, \phi}{E} &= \dpn{\text{Re}(x), \phi}{E} + \underbrace{i\dpn{\text{Im}(x), \phi}{E}}_{\in \real} = +\dpn{\text{Re}(x), \phi}{E} \\ +&\le \norm{\text{Re}(x)}_E \cdot \sup\bracsn{\dpn{y, \phi}{E}|y \in E, y = y^*, \norm{y}_E = 1} + \end{align*} + + where $\norm{\text{Re}(x)}_E = \norm{{(x + x^*)}/{2}}_E \le \norm{x}_E$. As the above holds for all $x \in E$, + \[ + \norm{\phi}_{E^*} \le \sup\bracsn{\dpn{x, \phi}{E}|x \in E, x = x^*, \norm{x}_E = 1} + \] +\end{proof} + diff --git a/src/op/banach/spectrum.tex b/src/op/banach/spectrum.tex index 4559c2d..f982f5f 100644 --- a/src/op/banach/spectrum.tex +++ b/src/op/banach/spectrum.tex @@ -128,7 +128,7 @@ \begin{proposition} \label{proposition:commutative-spectrum-gymnastics} - Let $A$ be a commutative unital Banach algebra and $x, y \in A$ with $x = y$, then + Let $A$ be a commutative unital Banach algebra and $x, y \in A$ with $xy = yx$, then \begin{enumerate} \item $\sigma_A(x + y) \subset \sigma_A(x) + \sigma_A(y)$. \item $\sigma_A(xy) \subset \sigma_A(x)\sigma_A(y)$. @@ -161,7 +161,7 @@ (2): Let $\lambda \in \partial \sigma_B(x)$, then there exists $\seq{\lambda_n} \subset \complex \setminus \sigma_B(x)$ such that $\lambda_n - x \in G(B)$ for all $n \in \natp$, and $\lambda_n \to \lambda$ as $n \to \infty$. By \autoref{corollary:invertible-boundary-explode}, $\norm{(\lambda_n - x)^{-1}}_A \to \infty$ as $n \to \infty$. If $\lambda - x \in G(A)$, then $(\lambda_n - x)^{-1} \to (\lambda - x)^{-1}$ as $n \to \infty$. Thus $\norm{(\lambda - x)^{-1}}_A = \infty$, which is impossible. Therefore $\lambda - x \not\in G(A)$, and $\lambda \in \sigma_A(x)$. \end{proof} -\begin{theorem}["Runge's Theorem"] +\begin{theorem}[Runge] \label{theorem:spectrum-subalgebra-sufficiency} Let $A$ be a unital Banach algebra, $x \in A$, $P \subset \complex \setminus \sigma_A(x)$ such that $P$ intersects every bounded component of $\complex \setminus \sigma_A(x)$, and $B \subset A$ be a closed algebra containing $1$, $x$, and $\bracsn{(\lambda - x)^{-1}|\lambda \in P}$, then $\sigma_A(x) = \sigma_B(x)$. \end{theorem} diff --git a/src/op/c-star/index.tex b/src/op/c-star/index.tex index d89067e..bcf7b55 100644 --- a/src/op/c-star/index.tex +++ b/src/op/c-star/index.tex @@ -8,4 +8,5 @@ \input{./homomorphism.tex} \input{./gelfand.tex} \input{./cont.tex} -\input{./order.tex} \ No newline at end of file +\input{./order.tex} +\input{./positive.tex} \ No newline at end of file diff --git a/src/op/c-star/positive.tex b/src/op/c-star/positive.tex new file mode 100644 index 0000000..7ee408b --- /dev/null +++ b/src/op/c-star/positive.tex @@ -0,0 +1,31 @@ +\section{Positive Linear Functionals} +\label{section:cstar-positive} + +\begin{definition}[Positive Linear Functional] +\label{definition:cstar-positive-functional} + Let $A$ be a unital $C^*$-algebra and $\phi \in A^*$, then $\phi$ is \textbf{positive} if $\dpn{x, \phi}{A} \ge 0$ for all positive elements $x \in A$. +\end{definition} + +\begin{theorem} +\label{theorem:cstar-positive-algebraic} + Let $A$ be a unital $C^*$-algebra and $\phi \in \hom(A; \complex)$, then the following are equivalent: + \begin{enumerate} + \item $\phi$ is a positive linear functional. + \item $\phi \in A^*$ with $\normn{\phi}_{A^*} = \dpn{1, \phi}{A}$. + \end{enumerate} +\end{theorem} +\begin{proof} + (1) $\Rightarrow$ (2): For any $x \in A_{sa}$ with $\norm{x}_A \le 1$, $\sigma_A(1 - x) \subset 1 - [-1, 1] = [0, 2]$ by \autoref{proposition:commutative-spectrum-gymnastics}. Thus $1 - x \ge 0$ by \autoref{proposition:positive-spectrum}, and $\dpn{x, \phi}{A} \le \dpn{1, \phi}{A}$. By \autoref{proposition:hermitian-functional-norm}, $\norm{\phi}_{A^*} \le \dpn{1, \phi}{A}$, so $\norm{\phi}_{A^*} = \dpn{1, \phi}{A}$. + + (2) $\Rightarrow$ (1): For each $x \in A_{sa}$, by restricting to $A[x]$, assume without loss of generality that $A$ is commutative. By the \autoref{theorem:riesz-radon-c0}, $\phi$ takes the form of a Radon measure $\mu$ on $\Omega(A)$, and $\norm{\phi}_{A^*} = \norm{\mu}_{\text{var}}$. For each Borel set $E \in \cb_{\Omega(A)}$, + \begin{align*} + \norm{\mu}_{\text{var}} &= \int_{\Omega(A)} 1 d\mu = \mu(E) + \mu(\Omega(A) \setminus E) \\ + &\le |\mu(E)| + |\mu(\Omega(A) \setminus E)| \le \norm{\mu}_{\text{var}} + \end{align*} + + which is only possible if $\mu(E), \mu(\Omega(A) \setminus E) \ge 0$. As this holds for all $E \in \cb_{\Omega(A)}$, $\mu$ is positive, and $\phi$ then is a positive linear functional. + +\end{proof} + + + diff --git a/src/op/c-star/sa.tex b/src/op/c-star/sa.tex index 529e74c..b0d142a 100644 --- a/src/op/c-star/sa.tex +++ b/src/op/c-star/sa.tex @@ -6,7 +6,7 @@ Let $A$ be an involutive algebra over $\complex$ and $x \in A$, then $x$ is \textbf{self-adjoint} if $x = x^*$. The space $A_{sa} = \bracs{x \in A| x = x^*}$ is the \textbf{self-adjoint part} of $A$, and: \begin{enumerate} \item $A_{sa}$ is a $\real$ subspace of $A$. - \item $A = \complex(A_{sa})$ as a vector space. + \item $A = \complex(A_{sa})$, with equivalent norms. \item For each $x \in A$, let \[ \text{Re}(x) = \frac{x + x^*}{2} \quad \text{Im}(x) = \frac{x - x^*}{2i} @@ -64,7 +64,7 @@ \end{enumerate} \end{corollary} \begin{proof} - (1): Since $\sigma_A(x)$ is compact, there exisst $\lambda \in \sigma_A(x)$ such that $|\lambda| = [x]_{sp}$. By \autoref{theorem:c-star-normal-spectral-radius}, $|\lambda| = [x]_{sp} = \norm{x}_A$. + (1): Since $\sigma_A(x)$ is compact, there exists $\lambda \in \sigma_A(x)$ such that $|\lambda| = [x]_{sp}$. By \autoref{theorem:c-star-normal-spectral-radius}, $|\lambda| = [x]_{sp} = \norm{x}_A$. (2): By the \hyperref[Spectral Mapping Theorem]{theorem:spectral-mapping-holomorphic}, \[