Corollary

Let A be a commutative involutive Banach algebra and $j$ the canonical morphism of A into Stell(A). The mapping $\mathsf{X}(j)$ is a homeomorphism of $\mathsf{X}$(Stell(A)) onto the subspace $\mathsf{X}(A)_h$ of $\mathsf{X}(A)$ formed by the Hermitian characters of A.

The Hermitian characters of A are the morphisms of involutive algebras from A into the stellar algebra $\mathbf{C}$. Prop. 20 therefore entails that $\mathsf{X}(j)$ is a bijection of $\mathsf{X}$(Stell(A)) onto $\mathsf{X}(A)_h$. Since $\mathsf{X}(j)$ is a homeomorphism onto its image (cf. I, p. 10), the corollary follows.

We identify $\mathsf{X}$(Stell(A)) with $\mathsf{X}(A)_h$ by means of the mapping $\mathsf{X}(j)$. For every $x\in$ A, the mapping $\mathscr{G}_{Stell(A)}(j(x))$ is none other than the restriction of $\mathscr{G}_A(x)$ to $\mathsf{X}(A)_h$.