\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}