Given an unital commutative c-star algebra $\mathcal A$ the Gelfand spectrum is the set $\Sigma(\mathcal A)$ of multiplicative linear functionals (i.e., continuous nonzero linear homomorphisms $\mathcal A \to \mathbb C$) together with a canonical topology called spectral topology, and which is compact Hausdorff. Multiplicative linear functionals are also called characters of $\mathcal{A}$.
There is a 1-1 relation between $\Sigma(\mathcal A)$ and the set of proper maximal ideals of $\mathcal A$ (see @strocchi2008introduction page 27, prop 1.5.3). So following the spirit of Algebraic Geometry, it can be interpreted as the points of a space for which $\mathcal A$ is the set of complex-valued continuous functions. It reminds me the spectrum of a ring.
It can be also defined the spectrum of an element $f\in \mathcal A$ as
$$ \sigma(f)=\{\lambda \in \mathbb C:f-\lambda \mbox{id} \text{ is not invertible in } \mathcal A\}. $$When $\mathcal A$ is generated by a single element $A$, i.e. the linear span of the powers of $A$ is dense in $\mathcal A$, then
$$ \Sigma(\mathcal A)=\sigma(A) $$(@strocchi2008introduction page 27).
Moreover, if $\mathcal A$ is generated by the algebraically independent elements $A_1,\ldots,A_n,A_1^*,\ldots,A_n^*$ then
$$ \Sigma(\mathcal A)=\sigma(A_1)\times \cdots \times \sigma(A_n). $$________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: