Example 1

Let A be the algebra of complex-valued functions on a set X. The mapping $f\mapsto \overline{f}$ is an involution in A. The subalgebra of bounded functions on X is an involutive subalgebra of A. If X is a locally compact topological space, the subalgebras $\mathscr{C}(X),\mathscr{C}_b(X),\mathscr{C}_0(X)$ and $\mathscr{K}(X)$ are involutive subalgebras of A.