From 09a94756ea7178d74017cd74575c2f8d82179328 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Thu, 4 Jun 2026 17:54:06 -0400 Subject: [PATCH] Added basics of C*-algebras. --- src/fa/tvs/complexify.tex | 5 ++-- src/op/c-star/index.tex | 6 ++++ src/op/c-star/involution.tex | 57 ++++++++++++++++++++++++++++++++++++ src/op/c-star/order.tex | 10 +++++++ src/op/c-star/sa.tex | 36 +++++++++++++++++++++++ src/op/index.tex | 1 + 6 files changed, 113 insertions(+), 2 deletions(-) create mode 100644 src/op/c-star/index.tex create mode 100644 src/op/c-star/involution.tex create mode 100644 src/op/c-star/order.tex create mode 100644 src/op/c-star/sa.tex diff --git a/src/fa/tvs/complexify.tex b/src/fa/tvs/complexify.tex index e571628..bd00c2b 100644 --- a/src/fa/tvs/complexify.tex +++ b/src/fa/tvs/complexify.tex @@ -58,8 +58,8 @@ \label{definition:complex-conjugation} Let $E$ be a vector space over $\complex$ and $*: E \to E$ be a $\real$-linear map, then $*$ is a \textbf{complex conjugation} if: \begin{enumerate} - \item For each $\lambda \in \complex$, $(\lambda x)^* = \ol \lambda x^*$. - \item For each $x \in E$, $x^{**} = x$. + \item[(C1)] For each $\lambda \in \complex$, $(\lambda x)^* = \ol \lambda x^*$. + \item[(C2)] For each $x \in E$, $x^{**} = x$. \end{enumerate} In which case, $\text{Re}(E) = \bracs{x \in E| x^* = x}$ is the \textbf{real part} of $E$. @@ -74,6 +74,7 @@ \[ \text{Re}(x) = \frac{x + x^*}{2} \quad \text{Im}(x) = \frac{x - x^*}{2i} \] + \item For each $x \in E$, $x^* = \text{Re}(x) - i\text{Im}(x)$. \end{enumerate} \end{proposition} \begin{proof} diff --git a/src/op/c-star/index.tex b/src/op/c-star/index.tex new file mode 100644 index 0000000..bd63399 --- /dev/null +++ b/src/op/c-star/index.tex @@ -0,0 +1,6 @@ +\chapter{$C^*$-Algebras} +\label{chap:c-star-algebras} + +\input{./involution.tex} +\input{./sa.tex} +\input{./order.tex} diff --git a/src/op/c-star/involution.tex b/src/op/c-star/involution.tex new file mode 100644 index 0000000..83219c0 --- /dev/null +++ b/src/op/c-star/involution.tex @@ -0,0 +1,57 @@ +\section{Involutions} +\label{section:involutions} + +\begin{definition}[Involution] +\label{definition:involution} + Let $A$ be an associative algebra over $\complex$, and $*: A \to A$ be a $\real$-linear map, then $*$ is an \textbf{involution} if: + \begin{enumerate} + \item[(C1)] For each $\lambda \in \complex$, $(\lambda x)^* = \ol \lambda x^*$. + \item[(C2)] For each $x \in A$, $x^{**} = x$. + \item[(I)] For every $x, y \in A$, $(xy)^* = y^*x^*$. + \end{enumerate} + + The space $A$ equipped with an involution is an \textbf{involutive algebra} over $\complex$. +\end{definition} + +\begin{definition}[$C^*$-Algebra] +\label{definition:c-star-algebra} + Let $A$ be an involutive Banach algebra over $\complex$, then $A$ is a \textbf{$C^*$-algebra} if for every $x \in A$, $\normn{x^*x}_A = \norm{x}_A^2$. +\end{definition} + +\begin{proposition} +\label{proposition:c-star-algebra-gymnastics} + Let $A$ be a $C^*$ algebra, then: + \begin{enumerate} + \item For each $x \in A$, $\norm{x}_A = \normn{x^*}_A\norm{x}_A$. + \end{enumerate} + + If $A$ is unital, then + \begin{enumerate}[start=1] + \item For each $\lambda \in \complex$, $\lambda^* = \ol \lambda$. + \item For any $x \in A$, $x \in G(A)$ if and only if $x^* \in G(A)$. + \item For every $x \in A$, $\sigma_A(x^*) = \bracsn{\ol \lambda| \lambda \in \sigma_A(x)}$. + \item For each $x \in A$, $[x]_{sp} = [x^*]_{sp}$. + \end{enumerate} +\end{proposition} +\begin{proof} + (1): For each $x \in A$, $\norm{x}_A^2 = \normn{x^*x}_A \le \norm{x}_A \normn{x^*}_A$. + + (2): For every $x \in A$, $1^*x^* = (x1)^* = x^* = (1x)^* = x^*1^*$, so $1^* = 1$ by uniqueness of the inverse. + + (3): For any $x \in A$, $(x^{-1})^*x^* = (x^{-1}x)^* = 1 = (xx^{-1})^* = x^*(x^{-1})^*$. +\end{proof} + +\begin{definition}[*-Homomorphism] +\label{definition:star-homomorphism} + Let $A, B$ be $C^*$-algebras and $\phi: A \to B$, then $\phi$ is a \textbf{*-homomorphism} if: + \begin{enumerate} + \item $\phi$ is a homomorphism of Banach algebras. + \item For every $x \in A$, $\phi(x^*) = \phi(x)^*$. + \end{enumerate} + + If in addition, $\phi(1) = 1$, then $\phi$ is a \textbf{unital *-homomorphism}. +\end{definition} + + + + diff --git a/src/op/c-star/order.tex b/src/op/c-star/order.tex new file mode 100644 index 0000000..706670e --- /dev/null +++ b/src/op/c-star/order.tex @@ -0,0 +1,10 @@ +\section{Order Structures of $C^*$-Algebras} +\label{section:order-c-star-algebra} + +\begin{definition}[Positive] +\label{definition:positive-c-star-algebra} + Let $A$ be a $C^*$-algebra and $x \in A$, then $x$ is \textbf{positive} if there exists $y \in A$ such that $x = y^*y$. +\end{definition} + + + diff --git a/src/op/c-star/sa.tex b/src/op/c-star/sa.tex new file mode 100644 index 0000000..7bb1478 --- /dev/null +++ b/src/op/c-star/sa.tex @@ -0,0 +1,36 @@ +\section{Self-Adjoint Elements} +\label{section:c-star-self-adjoint} + +\begin{definition}[Self-Adjoint] +\label{definition:self-adjoint} + 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 For each $x \in A$, let + \[ + \text{Re}(x) = \frac{x + x^*}{2} \quad \text{Im}(x) = \frac{x - x^*}{2i} + \] + + then $\text{Re}(x), \text{Im}(x) \in A_{sa}^2$ and $x = \text{Re}(x) + i\text{Im}(x)$. + \item For each $x \in A$, $x^* = \text{Re}(x) - i\text{Im}(x)$. + \end{enumerate} +\end{definition} +\begin{proof} + By \autoref{proposition:complex-conjugation-properties}. +\end{proof} + +\begin{definition}[Normal] +\label{definition:c-star-normal} + Let $A$ be an involutive algebra over $\complex$ and $x \in A$, then the following are equivalent: + \begin{enumerate} + \item $\text{Re}(x)\text{Im}(x) = \text{Im}(x)\text{Re}(x)$. + \item $x^*x = xx^*$. + \end{enumerate} + + If the above holds, then $x$ is \textbf{normal}. +\end{definition} + + + + diff --git a/src/op/index.tex b/src/op/index.tex index c7ed7e6..9fed099 100644 --- a/src/op/index.tex +++ b/src/op/index.tex @@ -2,5 +2,6 @@ \label{part:operator-algebras} \input{./banach/index.tex} +\input{./c-star/index.tex} \input{./example/index.tex} \input{./notation.tex} \ No newline at end of file