32 lines
2.6 KiB
TeX
32 lines
2.6 KiB
TeX
\chapter{Notations}
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\label{chap:op-notations}
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\begin{tabular}{lll}
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\textbf{Notation} & \textbf{Description} & \textbf{Source} \\
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\hline
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$1$ & Identity element of a unital algebra. & \autoref{definition:unital-banach-algebra} \\
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$Z(A)$ & Centre of a Banach algebra. & \autoref{definition:banach-algebra-centre} \\
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$G(A)$ & Invertible group of a unital algebra. & \autoref{definition:banach-algebra-invertible} \\
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$U(A)$ & Unitary group of a unital algebra. & \autoref{definition:banach-algebra-invertible} \\
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$G_0(A)$ & The identity component of $G(A)$. & \autoref{definition:identity-component} \\
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$I(A)$ & The index group of $A$. & \autoref{definition:index-group} \\
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$\sigma_A(x) = \sigma(x)$ & The spectrum of $x$ in $A$. & \autoref{definition:spectrum} \\
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$R_x(\lambda)$ & The resolvent of $x$. & \autoref{definition:resolvent} \\
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$[x]_{sp}$ & The spectral radius of $x$. & \autoref{definition:spectral-radius} \\
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$\Omega(A)$ & Space of multiplicative functionals on $A$. & \autoref{definition:multiplicative-functional} \\
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$\cm(A)$ & Maximal ideal space of $A$. & \autoref{definition:maximal-ideal} \\
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$\Gamma = \Gamma_A$ & The Gelfand transform on $A$. & \autoref{definition:gelfand-transform} \\
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$A[S]$ & $C^*$-subalgebra of $A$ generated by $S \subset A$. & \autoref{definition:generated-subalgebra} \\
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$S(A)$ & State space of a $C^*$-algebra $A$. & \autoref{definition:cstar-state} \\
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$PS(A)$ & Pure state space of a $C^*$-algebra $A$. & \autoref{definition:pure-state} \\
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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} \\
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$(H_\phi, \pi_\phi, \xi_\phi)$ & GNS triple associated with $\phi \in S(A)$. & \autoref{definition:gns-triple} \\
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$L^2(M, \phi)$ & Hilbert space of the GNS triple associated with a faithful $\phi \in S(A)$. & \autoref{definition:gns-triple} \\
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$M_n(\complex)$ & Algebra of $n \times n$ matrices over $\complex$. & \autoref{definition:matrix-algebra} \\
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$B(H)$ & Algebra of bounded operators on a Hilbert space. & \autoref{definition:hilbert-endomorphism} \\
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$A(D)$ & The disk algebra. & \autoref{definition:disk-algebra} \\
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$H^\infty(D)$ & The Hardy space. & \autoref{definition:hardy-space} \\
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$\ell^1(\integer)$ & Convolution algebra on $\integer$. &\autoref{definition:convolution-algebra-integer} \\
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$\delta_0$ & Multiplicative unit of $\ell^1(\integer)$. & \autoref{definition:convolution-algebra-integer} \\
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\end{tabular} |