Notation management.

src/op/notation.tex

@@ -7,6 +7,7 @@
     $1$ & Identity element of a unital algebra. & \autoref{definition:unital-banach-algebra} \\
     $Z(A)$ & Centre of a Banach algebra. & \autoref{definition:banach-algebra-centre} \\ 
     $G(A)$ & Invertible group of a unital algebra. & \autoref{definition:banach-algebra-invertible} \\
+    $U(A)$ & Unitary 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} \\
@@ -21,13 +22,7 @@
     $\dpn{x, y}{\phi}$ & Defined as $\dpn{y^*x, \phi}{A}$, the pseudo inner product associated to a positive linear functional. & \autoref{definition:cstar-state-pseudo-inner-product} \\ 
     $(H_\phi, \pi_\phi, \xi_\phi)$ & GNS triple associated with $\phi \in S(A)$. & \autoref{definition:gns-triple} \\ 
     $L^2(M, \phi)$ & Hilbert space of the GNS triple associated with a faithful $\phi \in S(A)$. & \autoref{definition:gns-triple} \\ 
-    $U(T)$ & 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} \\ 
-
+    
     $M_n(\complex)$ & 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} \\

src/vna/notation.tex

@@ -5,9 +5,37 @@
     \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}