Consider A = A(D)¹. Then f^*(z) = [`(f([`z]))] gives an involution on A such that ||f|| = sup_{z Î D}|f(z)| = sup_{z Î D}|f([`z])| = ||f^*||. Consider f(z) = z² and g(z) = z, then g is self-adjoint and f = gg^*. So f is positive and we must have sp(f) Í [0,¥) contradicting sp(f) = D. Hence A isn't a C^*-algebra.

Footnotes:

¹Let D denote the closed unit disc {z Î C, |z| £ 1}. Suppose that A(D) denoted the set of all elements of C(D) which are analytic on the interior of D. A(D) is a closed subalgebra of C(D).