Lemma 10

The mapping ev$_x:\chi \mapsto \chi (x)$ induces a homeomorphism of $\mathsf{X}(B)$ onto Sp$_A(x)$.

The mapping $x\mapsto \chi (x)$ of $\mathsf{X}(B)$ into $\mathbf{C}$ is continuous, and its image is equal to Sp$_B(x)$ by prop. 6 of I, p. 37, hence to Sp$_A(x)$. Since the characters of B are Hermitian (lemma 9 of I, p. 107), characters of B which coincide at $x$ are also equal at $x^*$, hence are equal on the unital star-subalgebra B generated by $x$. This proves that the mapping ev$_x$ is injective. Since $\mathsf{X}(B)$ is compact and $\mathbf{C}$ is separated, it induces a homeomorphism of $\mathsf{X}(B)$ onto its image, whence the lemma.

From the lemma one deduces an isomorphism of star-algebras $\varphi_x:\mathscr{C}$(Sp$_A(x)$)$\rightarrow \mathscr{C}(\mathsf{X}(B))$. It carries a function $f\in \mathscr{C}$ (Sp$_A(x)$) to the function $\chi \mapsto f(\chi (x))$. The mapping $\mathscr{G}_B^{-1}\circ \varphi_x$ is an isomorphism of the star-algebra $\mathscr{C}$(Sp$_A(x)$) onto B.