An involutive Banach algebra A which isn’t a C^{∗}-algebra.

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Consider A = A(Δ)^{1}. Then f^{∗}(z) = \overline{f(z̄)} gives an involution on A such that ∥f∥ = sup_{z∈D} |f(z)| = sup_{z∈D} |f(z̄)| = ∥f^{∗}∥. Consider f(z) = z^{2} and g(z) = z, then g is self-adjoint and f = gg^{∗}. So f is positive and we must have sp(f) ⊆ [0, ∞) contradicting sp(f) = Δ. Hence A isn’t a C^{∗}-algebra.

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