Gelfand spectrum

Given an unital commutative c-star algebra $\mathcal{A}$ the Gelfand spectrum is the set $\mathrm{\Sigma }(\mathcal{A})$ 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 $\mathrm{\Sigma }(\mathcal{A})$ 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

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

(@strocchi2008introduction page 27).

Moreover, if $\mathcal{A}$ is generated by the algebraically independent elements $A_1,\dots ,A_n,A_1^{\ast },\dots ,A_n^{\ast }$ then