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

\begin{tabular}{lll}
    \textbf{Notation} & \textbf{Description} & \textbf{Source} \\
    \hline
    
    
    $U(T)$ & Cayley transform of $T$. & \autoref{definition:cayley-transform-bounded} \\ 
    $E_{x, y}$ & Section of a spectral measure $E$, where $E_{x, y}(B) = \dpn{E(B)x, y}{H}$. & \autoref{definition:spectral-measure} \\ 
    $\text{Proj}(A)$ & Projections in $A$. & \autoref{definition:vn-projection-lattice} \\
    $Z(P)$ & Central support of $P \in \text{Proj}(A)$. & \autoref{definition:central-support-vna} \\ 
    $P \sim Q$ & $P, Q \in \text{Proj}(A)$ are Murray-von Neumann equivalent & \autoref{definition:murray-von-neumann-equivalent} \\ 
    $P \preceq Q$ & $P$ is Murray-von Neumann subequivalent to $Q$. & \autoref{definition:murray-von-neumann-subequivalent}

    $A'$ & Commutant of $A \subset B(H)$, the operators in $B(H)$ commuting with every element of $A$. & \autoref{definition:commutant} \\
    $I_E(\phi) = \int_X \phi dE$ & Integral of $\phi$ with respect to a spectral measure $E$. & \autoref{definition:spectral-measure-integral} \\
    $\phi(T)$ & $L^\infty$-functional calculus of a normal operator $T$, defined as $\int_{\sigma_{B(H)}(T)}\phi dE$. & \autoref{definition:linfty-functional-calculus} \\
    $A_{\vnI}, A_{\vnII_1}, A_{\vnII_\infty}, A_{\vnIII}$ & Type $\vnI$, $\vnII_1$, $\vnII_\infty$, and $\vnIII$ summands of a von Neumann algebra $A$. & \autoref{theorem:vna-type-decomposition} \\
 
    $\lambda$ & Left regular representation of $G$, and its extension to $L^1(G; \complex)$. & \autoref{definition:group-von-neumann-algebra} \\
    $\rho$ & Right regular representation of $G$. & \autoref{definition:group-von-neumann-algebra} \\
    $L(G)$ & Group von Neumann algebra of $G$, defined as $\lambda(G)''$. & \autoref{definition:group-von-neumann-algebra} \\
    $R(G)$ & Defined as $\rho(G)''$, the von Neumann algebra generated by the right regular representation. & \autoref{definition:group-von-neumann-algebra} \\
    $L_f$ & Left convolution map $g \mapsto f * g$ by a left convolver $f$. & \autoref{definition:left-convolver} \\
    $LC(G)$ & Algebra of left convolvers on $G$. & \autoref{definition:left-convolver-algebra} \\
    $f^*$ & Defined as $f^*(x) = \ol{f(x^{-1})}$ for a left convolver $f$. & \autoref{definition:left-convolver-algebra} \\
    $u_x$ & Canonical unitary $\lambda(x)$ of $L(G)$, identified with $\delta_x$. & \autoref{definition:group-algebra-fourier-series} \\
    $T_x$ & Fourier coefficients of $T \in L(G)$, where $T = \sum_{x \in G}T_xu_x$. & \autoref{definition:group-algebra-fourier-series} \\
    ICC & Infinite conjugacy classes: every non-trivial conjugacy class is infinite. & \autoref{definition:icc} \\
    $H^{(I)}$ & Defined as $\bigoplus_{i \in I}H$. & \autoref{definition:wreath-product} \\
    $H \wr_I \Gamma$ & Generalised wreath product $H^{(I)} \rtimes \Gamma$. & \autoref{definition:wreath-product} \\
    $H \wr \Gamma$ & Wreath product, the case $I = \Gamma$ with $\Gamma$ acting by left translation. & \autoref{definition:wreath-product} \\
    $\lim_{\longrightarrow}(M_i, \tau_i)$ & Inductive limit of an upward-directed system of tracial von Neumann algebras. & \autoref{definition:inductive-limit-tracial-von-neumann-algebra} \\
    $M_2(\complex)^{\ol \otimes \infty}$ & The hyperfinite $\vnII_1$ factor, defined as $\lim_{\longrightarrow}(M_{2^n}(\complex), \tau_n)$. & \autoref{definition:hyperfinite-ii1} \\
 
    $Mq$ & Induced von Neumann algebra of $M$ on $qH$ with respect to $q \in \text{Proj}(M')$. & \autoref{definition:induced-vna-commutant} \\
    $M^t$ & Amplification of a $\vnII_1$ factor $M$, defined as $pM_n(M)p$ with $(\text{Tr}_n \otimes \tau)p = t$. & \autoref{definition:ii1-amplification} \\
    $\fF(M)$ & Fundamental group of a $\vnII_1$ factor $M$, defined as $\bracsn{t > 0|M^t \iso M}$. & \autoref{definition:ii1-fundamental-group}

\end{tabular}