A nonclosed ideal that is not self-adjoint in a commutative C^{∗}-algebra.

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Consider C^{∗}-algebra A = C(Δ) and the ideal I = fA = {fg ; g ∈ A}, where f(z) = z. f^{∗}(z) = z̄ and if f^{∗} ∈ I, then there exists an element g ∈ A such that f^{∗} = fg. So g(0) = lim_{z→0} g(z) = lim_{z→0} \frac{z̄}{z}, a contradiction. Thus I isn’t self-adjoint.