A closed subalgebra of a C^{∗}-algebra that isn’t self-adjoint.

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The disc algebra A(Δ) ^{1} is a closed subalgebra of the C^{∗}-algebra C(Δ). If f and f̄ both belong to A(Δ), then by the Cauchy-Riemann equations f will be constant. So A(Δ) isn’t self-adjoint.

1(VIII) Let 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).