It is known (see [Mo88b]) that the axiom of choice is equivalent to the following statement: The closed unit ball of the continuous dual of a unitary commutative normed algebra has at least one extreme point.
In this paper we show that the two following axioms are equivalent:
UF: Every filter on a set is contained in a ultrafilter.
GE: The closed unit ball of the continuous dual of a non-trivial Gelfand algebra has at least one extreme point which is not 0.