42 Notations
| Notation | Description | Source |
| $1$ | Identity element of a unital algebra. | Definition 38.1.2 |
| $Z(A)$ | Centre of a Banach algebra. | Definition 38.1.3 |
| $G(A)$ | Invertible group of a unital algebra. | Definition 38.2.1 |
| $U(A)$ | Unitary group of a unital algebra. | Definition 38.2.1 |
| $G_{0}(A)$ | The identity component of $G(A)$. | Definition 38.3.1 |
| $I(A)$ | The index group of $A$. | Definition 38.3.3 |
| $\sigma_{A}(x) = \sigma(x)$ | The spectrum of $x$ in $A$. | Definition 38.5.1 |
| $R_{x}(\lambda)$ | The resolvent of $x$. | Definition 38.5.3 |
| $[x]_{sp}$ | The spectral radius of $x$. | Definition 38.5.2 |
| $\Omega(A)$ | Space of multiplicative functionals on $A$. | Definition 38.7.1 |
| $\cm(A)$ | Maximal ideal space of $A$. | Definition 38.4.3 |
| $\Gamma = \Gamma_{A}$ | The Gelfand transform on $A$. | Definition 38.8.1 |
| $A[S]$ | $C^{*}$-subalgebra of $A$ generated by $S \subset A$. | Definition 39.2.1 |
| $S(A)$ | State space of a $C^{*}$-algebra $A$. | Definition 39.10.1 |
| $PS(A)$ | Pure state space of a $C^{*}$-algebra $A$. | Definition 39.10.3 |
| $\dpn{x, y}{\phi}$ | Defined as $\dpn{y^*x, \phi}{A}$, the pseudo inner product associated to a positive linear functional. | Definition 39.10.2 |
| $(H_{\phi}, \pi_{\phi}, \xi_{\phi})$ | GNS triple associated with $\phi \in S(A)$. | Definition 39.11.3 |
| $L^{2}(M, \phi)$ | Hilbert space of the GNS triple associated with a faithful $\phi \in S(A)$. | Definition 39.11.3 |
$M_{n}(\complex)$ | Algebra of $n \times n$ matrices over $\complex$. | Definition 40.1.1 |
| $B(H)$ | Algebra of bounded operators on a Hilbert space. | Definition 40.2.1 |
| $A(D)$ | The disk algebra. | Definition 40.5.1 |
| $H^{\infty}(D)$ | The Hardy space. | Definition 40.4.1 |
| $\ell^{1}(\integer)$ | Convolution algebra on $\integer$. | Definition 40.6.1 |
| $\delta_{0}$ | Multiplicative unit of $\ell^{1}(\integer)$. | Definition 40.6.1 |
Post a Comment