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Notation management.

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Bokuan Li
2026-11-06 19:16:21 -05:00
parent 400382be1d
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src/op/notation.tex
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$1$ & 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} \\
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$\dpn{x, y}{\phi}$ & 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} \\