34 Notations
| Notation | Description | Source |
$1 = 1_{G}$ | Identity element of the group $G$. | Lemma 31.2.2 |
| $C_{c}^{+}(G)$ | Space of non-negative continuous compactly supported functions on $G$. | Definition 31.2.3 |
| $L_{x} f$ | Left translation of $f$ by $x$, $L_{x}f(y) = f(x^{-1}y)$. | Definition 31.2.3 |
| $R_{x} f$ | Right translation of $f$ by $x$, $R_{x}f(y) = f(yx)$. | Definition 31.3.1 |
| $(f : g)$ | Covering ratio of $f$ by $g$. | Definition 31.2.3 |
| $L^{p}(G; E)$ | $L^{p}(G, \mu; E)$, where $\mu$ is a fixed left Haar measure on $G$. | Definition 31.2.8 |
| $\int_{G} f(x) dx$ | Integral of $f$ with respect to the fixed left Haar measure. | Definition 31.2.8 |
| $|A|$ | Haar measure of a Borel set $A \subset G$. | Definition 31.2.8 |
| $\Delta_{G}$ | Modular function of $G$. | Definition 31.3.1 |
| $\mu * \nu$ | Convolution of finite Radon measures. | Definition 31.4.1 |
| $\mu^{*}$ | Involution of a measure, $\mu^{*}(A) = \overline{\mu(A^{-1})}$. | Definition 31.4.3 |
| $f * g$ | Convolution of functions on $G$. | Definition 31.4.4 |
| $f^{*}$ | Involution on $L^{1}(G; \complex)$, $f^{*}(x) = \Delta_{G}(x^{-1})\overline{f(x^{-1})}$. | Definition 31.4.7 |
| $\bigoplus_{i \in I}\pi_{i}$ | Direct sum of unitary representations. | Definition 32.1.7 |
| $\pi_{L}$ | Left regular representation of $G$. | Definition 32.1.2 |
| $\mathcal{C}(\pi_{1}, \pi_{2})$ | Space of intertwining operators between $\pi_{1}$ and $\pi_{2}$. | Definition 32.1.3 |
| $S(G)$ | State space of $G$. | Definition 32.3.2, Definition 32.4.2 |
| $\langle f, g \rangle_{\phi}$ | Pseudo inner product $\langle g^{*} * f, \phi \rangle_{L^1(G; \complex)}$ induced by a positive linear functional $\phi$. | Definition 32.3.3 |
| $(H_{\phi}, \pi_{\phi}, \xi_{\phi})$ | GNS triple associated with $\phi$. | Definition 32.3.3 |
| $P(G)$ | Space of positive-definite functions on $G$. | Definition 32.4.2 |
| $P_{0}(G)$ | Space of positive-definite functions of norm at most $1$. | Definition 32.4.2 |
| $\mathbb{T}$ | Unit circle $\partial B_{\complex}(0, 1)$. | Definition 33.1.1 |
| $\widehat{G}$ | Dual group of a locally compact Abelian group $G$. | Definition 33.1.1 |
| $\langle x, \phi \rangle_{G}$ | Dual pairing, $\langle x, \phi \rangle_{G} = \phi(x)$ for $\phi \in \widehat{G}$. | Definition 33.1.1 |
| $\pi_{\phi}$ | Representation of $L^{1}(G; \complex)$ determined by the character $\phi \in \widehat{G}$. | Theorem 33.1.2 |
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