### Amenability for discrete convolution semigroup algebras.

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There was a standard procedure in Mesopotamia for solving quadratic problems involving lengths and areas of squares. In this paper, we show, by means of an example from Susa, how area constants were used to reduce problems involving other geometrical figures to the standard form.

Given ⊂ ℕ, let ̂ be the set of all positive integers m for which ${h}^{m}$ is hermitian whenever h is an element of a complex unital Banach algebra A with hⁿ hermitian for each n ∈ . We attempt to characterize when (i) ̂ = ℕ, or (ii) ̂ = . A key tool is a Müntz-type theorem which gives remarkable conclusions when 1 ∈ and ∑ 1/n: n ∈ diverges. The set ̂ is determined by a single extremal Banach algebra Ea(). We describe this extremal algebra for various .

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