40 Notations
| Notation | Description | Source |
| $1$ | Identity element of a unital algebra. | Definition 36.1.2 |
| $Z(A)$ | Centre of a Banach algebra. | Definition 36.1.3 |
| $G(A)$ | Invertible group of a unital algebra. | Definition 36.2.1 |
| $G_{0}(A)$ | The identity component of $G(A)$. | Definition 36.3.1 |
| $I(A)$ | The index group of $A$. | Definition 36.3.3 |
| $\sigma_{A}(x) = \sigma(x)$ | The spectrum of $x$ in $A$. | Definition 36.5.1 |
| $R_{x}(\lambda)$ | The resolvent of $x$. | Definition 36.5.3 |
| $[x]_{sp}$ | The spectral radius of $x$. | Definition 36.5.2 |
| $\Omega(A)$ | Space of multiplicative functionals on $A$. | Definition 36.7.1 |
| $\cm(A)$ | Maximal ideal space of $A$. | Definition 36.4.3 |
| $\Gamma = \Gamma_{A}$ | The Gelfand transform on $A$. | Definition 36.8.1 |
| $A[S]$ | $C^{*}$-subalgebra of $A$ generated by $S \subset A$. | Definition 37.2.1 |
| $S(A)$ | State space of a $C^{*}$-algebra $A$. | Definition 37.10.1 |
| $PS(A)$ | Pure state space of a $C^{*}$-algebra $A$. | Definition 37.10.3 |
| $\dpn{x, y}{\phi}$ | Defined as $\dpn{y^*x, \phi}{A}$, the pseudo inner product associated to a positive linear functional. | Definition 37.10.2 |
| $(H_{\phi}, \pi_{\phi}, \xi_{\phi})$ | GNS triple associated with $\phi \in S(A)$. | Definition 37.11.3 |
| $U(T)$ | Cayley transform of $T$. | Definition 38.2.1 |
| $E_{x, y}$ | $E_{x, y}(B) = \dpn{E(B)x, y}{H}$. | Definition 38.5.1 |
| $\text{Proj}(A)$ | Projections in $A$. | Definition 38.7.1 |
| $Z(P)$ | Central support of $P \in \text{Proj}(A)$. | Definition 38.7.2 |
| $P \sim Q$ | $P, Q \in \text{Proj}(A)$ are Murray-von Neumann equivalent | Definition 38.7.8 |
| $P \preceq Q$ | $P$ is Murray-von Neumann subequivalent to $Q$. | Definition 38.7.9 |
$M_{n}(\complex)$ | Algebra of $n \times n$ matrices over $\complex$. | Definition 39.1.1 |
| $B(H)$ | Algebra of bounded operators on a Hilbert space. | Definition 39.2.1 |
| $A(D)$ | The disk algebra. | Definition 39.5.1 |
| $H^{\infty}(D)$ | The Hardy space. | Definition 39.4.1 |
| $\ell^{1}(\integer)$ | Convolution algebra on $\integer$. | Definition 39.6.1 |
| $\delta_{0}$ | Multiplicative unit of $\ell^{1}(\integer)$. | Definition 39.6.1 |
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