Example 2

Let X be a locally compact topological space. The Banach involutive algebra $\mathscr{C}_b(X)$ of continuous bounded complex-valued functions on X (example 1 of I, p. 100) is a stellar algebra. In fact, for every function $f\in \mathscr{C}_b(X)$, one has $f^*f=|f|^2$, and therefore $\|f^*f\|=\||f|^2\|=\|f\|^2$.

Let $A =\mathscr{C}_0(X)$ be the Banach involutive subalgebra of functions tending to 0 at infinity. It is a stellar subalgebra of $\mathscr{C}_b(X)$. For every function $f\in A$, one has $\|f\|=\varrho (f)$, since Sp$'_A(f) =f(X)\cup \{0\}$.

Let X and Y be locally compact topological spaces. For every proper partial mapping $\varphi$ from X into Y (def. 1 of I, p. 33), the algebra morphism $\varphi^*$ from $\mathscr{C}_0(Y)$ into $\mathscr{C}_0(X)$ (prop. 3 of I, p. 34) is a morphism of involutive algebras. Conversely, every morphism of stellar algebras $\pi :\mathscr{C}_0(Y)\rightarrow \mathscr{C}_0(X)$ is of this form (loc. cit.).