Geodesic
Interpolation and inequalities for functions of exponential type: the Arens irregularity of an extremal algebra
Glasgow mathematical journal, Tome 35 (1993) no. 3, pp. 325-326

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For any compact convex set K ⊂ C there is a unital Banach algebra Ea(K) generated by an element h in which every polynomial in h attains its maximum norm over all Banach algebras subject to the numerical range V(h) being contained in K, [1]. In the case of K a line segment in R, we show here that Ea(K) does not have Arens regular multiplication. We also show that ideas about Ea(K) give simple proofs of, and extend, two inequalities of C. Frappier [4]. 
Crabb, M. J.; McGregor, C. M. Interpolation and inequalities for functions of exponential type: the Arens irregularity of an extremal algebra. Glasgow mathematical journal, Tome 35 (1993) no. 3, pp. 325-326. doi: 10.1017/S0017089500009897
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