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Extreme points, unitaries and invertible elements.

The set of all extreme points of the closed unit ball of a unital C*-algebra A was identified by R.V. Kadison as the maximal partial isometries in A. Let u be an element in A, it is said that u is unitary if uu* = u*u = 1_A, that is, if u is invertible in A with u^-11 = u*. It is clear that any unitary is an extreme point in A, however the reciprocal statement is not in general true.In 1991 G.K. Pedersen provided a characterisation of the unitary elements of a unital C*-algebra among the extreme points of its closed unit ball in termsof the distance to the group of invertible elements. We present an alternative proof of this result based on the different natures coexisting in a unital C*-algebra. It is a nice example of how to combine all the knowledge existing in the setting of associative and Jordan algebras with the more general triple theory.