Example 1

Let X be a locally compact topological space. The Banach algebra $\mathscr{C}_b(X)$ of complex-valued continuous bounded functions on X, endowed with the norm $\|f\|$ = sup$_{x\in X}|f(x)|$ and the involution $f\mapsto \overline{f}$, is an involutive Banach algebra. The subalgebra $\mathscr{C}_0(X)$ of continuous functions tending to 0 at infinity is a closed involutive subalgebra of $\mathscr{C}_b(X)$.