A reflexive Banach algebra whose dual is also a Banach algebra.

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The Banach algebra l^{p}^{1}, 1 < p < ∞ has the conjugate l^{q}, q = \frac{p}{p − 1}, in addition (l^{q})^{#} = l^{p}.

1Let (Ω, μ) be a measure space and L^{p}(Ω, μ) for 1 ≤ p < ∞ be the set of all complex valued measurable functions f on Ω (we assume f is equal to g if f = g a.e.[μ]) for which ∥f∥_{p} = (∫_{Ω} |f|^{p}dμ)^{\frac{1}{p}} < ∞. L^{p}(Ω, μ) with the norm ∥.∥_{p} is a Banach space and is a Hilbert space iff p = 2. L^{p}(Ω, μ) denoted by l^{p}(Ω) if μ is counting measure. In particular, l^{p}(N) denoted by l^{p}. If 1 ≤ p < ∞, then l^{p} can be regarded as a commutative Banach algebra with coordinatewise multiplication. (For p > 1, ∥fg∥_{p} ≤ ∥f∥_{p}∥g∥_{p} is a conclusion of Hőlder inequality.) The l^{p}, 1 ≤ p < ∞, with the involution f ⟼ \overline{f} is an involutive Banach algebra.