We consider the real Banach spaces H(A) of all hermitian elements of a complex Banach algebra A. We prove that if an even power of a \in N(A) is hermitian, then a is an extreme point of the unit ball of H(A) if and only if a^2 = 1. Moreover, if an odd power of a \in H(A) is hermitian and a is an extreme point of the unit ball of H(A), then a^3 = a.
HERZOG, GERD and SCHMOEGER, CHRISTOPH (2007) "Extreme Points of Certain Subsets of Hermitian Elements in Banach Algebras," Turkish Journal of Mathematics: Vol. 31: No. 2, Article 4. Available at: https://journals.tubitak.gov.tr/math/vol31/iss2/4