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Collected notations. Thank you Claude.

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Bokuan Li
2026-10-02 16:01:47 -04:00
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\chapter{Notations}
\label{chap:harmonic-notations}
\begin{tabular}{lll}
\textbf{Notation} & \textbf{Description} & \textbf{Source} \\
\hline
% ---- Locally Compact Groups ----
$1 = 1_G$ & Identity element of the group $G$. & \autoref{lemma:lc-sigma-compact} \\
$C_c^+(G)$ & Space of non-negative continuous compactly supported functions on $G$. & \autoref{definition:lcg-covering-ratio} \\
$L_x f$ & Left translation of $f$ by $x$, $L_xf(y) = f(x^{-1}y)$. & \autoref{definition:lcg-covering-ratio} \\
$R_x f$ & Right translation of $f$ by $x$, $R_xf(y) = f(yx)$. & \autoref{definition:modular-function} \\
% ---- Haar Measures ----
$(f : g)$ & Covering ratio of $f$ by $g$. & \autoref{definition:lcg-covering-ratio} \\
$L^p(G; E)$ & $L^p(G, \mu; E)$, where $\mu$ is a fixed left Haar measure on $G$. & \autoref{definition:haar-measure-lp-notation} \\
$\int_G f(x) dx$ & Integral of $f$ with respect to the fixed left Haar measure. & \autoref{definition:haar-measure-lp-notation} \\
$|A|$ & Haar measure of a Borel set $A \subset G$. & \autoref{definition:haar-measure-lp-notation} \\
% ---- Modular Function ----
$\Delta_G$ & Modular function of $G$. & \autoref{definition:modular-function} \\
% ---- Group Algebras ----
$\mu * \nu$ & Convolution of finite Radon measures. & \autoref{definition:lcg-convolution-measures} \\
$\mu^*$ & Involution of a measure, $\mu^*(A) = \overline{\mu(A^{-1})}$. & \autoref{definition:group-measure-algebra} \\
$f * g$ & Convolution of functions on $G$. & \autoref{definition:group-convolution-functions} \\
$f^*$ & Involution on $L^1(G; \complex)$, $f^*(x) = \Delta_G(x^{-1})\overline{f(x^{-1})}$. & \autoref{definition:l1-group-algebra} \\
% ---- Representations ----
$\bigoplus_{i \in I}\pi_i$ & Direct sum of unitary representations. & \autoref{definition:lcg-rep-direct-sum} \\
$\pi_L$ & Left regular representation of $G$. & \autoref{definition:lcg-left-regular} \\
$\mathcal{C}(\pi_1, \pi_2)$ & Space of intertwining operators between $\pi_1$ and $\pi_2$. & \autoref{definition:lcg-intertwining} \\
% ---- GNS Construction ----
$S(G)$ & State space of $G$. & \autoref{definition:group-l1-state}, \autoref{definition:group-positive-definite-function} \\
$\langle f, g \rangle_\phi$ & Pseudo inner product $\langle g^* * f, \phi \rangle_{L^1(G; \complex)}$ induced by a positive linear functional $\phi$. & \autoref{definition:gns-triple-lcg} \\
$(H_\phi, \pi_\phi, \xi_\phi)$ & GNS triple associated with $\phi$. & \autoref{definition:gns-triple-lcg} \\
% ---- Positive-Definite Functions ----
$P(G)$ & Space of positive-definite functions on $G$. & \autoref{definition:group-positive-definite-function} \\
$P_0(G)$ & Space of positive-definite functions of norm at most $1$. & \autoref{definition:group-positive-definite-function} \\
% ---- Locally Compact Abelian Groups ----
$\mathbb{T}$ & Unit circle $\partial B_\complex(0, 1)$. & \autoref{definition:unitary-character} \\
$\widehat{G}$ & Dual group of a locally compact Abelian group $G$. & \autoref{definition:unitary-character} \\
$\langle x, \phi \rangle_G$ & Dual pairing, $\langle x, \phi \rangle_G = \phi(x)$ for $\phi \in \widehat{G}$. & \autoref{definition:unitary-character} \\
$\pi_\phi$ & Representation of $L^1(G; \complex)$ determined by the character $\phi \in \widehat{G}$. & \autoref{theorem:unitary-character-isomorphism}
\end{tabular}