Geodesic
Some extremal algebras for hermitians
Glasgow mathematical journal, Tome 43 (2001) no. 1, pp. 29-38

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We study three extremal Banach algebras: (a) generated by two hermitian unitaries; (b) generated by an element of norm 1 all of whose odd positive powers are hermitian; (c) generated by an element of norm 1 all of whose even positive powers are hermitian. In all three cases the numerical range is found for various elements. The second algebra is shown to be isometrically isomorphic to a subalgebra of the first. The third algebra is identified with a space of functions. 
Crabb, M. J.; Duncan, J.; McGregor, C. M. Some extremal algebras for hermitians. Glasgow mathematical journal, Tome 43 (2001) no. 1, pp. 29-38. doi: 10.1017/S0017089501010047
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