\chapter{Notations}
\label{chap:op-notations}

\begin{tabular}{lll}
    \textbf{Notation} & \textbf{Description} & \textbf{Source} \\
    \hline
    $1$ & Identity element of a unital algebra. & \autoref{definition:unital-banach-algebra} \\
    $G(A)$ & Invertible group of a unital algebra. & \autoref{definition:banach-algebra-invertible} \\
    $G_0(A)$ & The identity component of $G(A)$. & \autoref{definition:identity-component} \\
    $I(A)$ & The index group of $A$. & \autoref{definition:index-group} \\
    $\sigma_A(x) = \sigma(x)$ & The spectrum of $x$ in $A$. & \autoref{definition:spectrum} \\
    $R_x(\lambda)$ & The resolvent of $x$. & \autoref{definition:resolvent} \\ 
    $[x]_{sp}$ & The spectral radius of $x$. & \autoref{definition:spectral-radius} \\
    $\Omega(A)$ & Space of multiplicative functionals on $A$. & \autoref{definition:multiplicative-functional} \\
    $\cm(A)$ & 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} \\
    $H^\infty(D)$ & The Hardy space.  & \autoref{definition:hardy-space} \\
    $\ell^1(\integer)$ & Convolution algebra on $\integer$. &\autoref{definition:convolution-algebra-integer} \\ 
    $\delta_0$ & Multiplicative unit of $\ell^1(\integer)$. & \autoref{definition:convolution-algebra-integer} \\ 
\end{tabular}