Characterizations of geometric tripotents in strongly facially symmetric spaces

The concept of a geometric tripotent is one of the key concepts in the theory of strongly facially symmetric spaces. This paper studies the properties of geometric tripotents. We establish necessary and sufficient conditions under which a norm-one element of the dual space (real or complex) of a strongly facially symmetric space is a geometric tripotent. We prove that two geometric tripotents in such a space are mutually orthogonal if and only if both their sum and difference have norm one. Furthermore, we show that the set of extreme points of the unit ball coincides with the set of maximal geometric tripotents in the dual of a strongly facially symmetric space. Finally, we examine the relationship between M-orthogonality and ordinary orthogonality in the dual of a complex strongly facially symmetric space, providing a geometric characterization of geometric tripotents.