Lattice of tripotents in a \(\text{JBW}^*\)-triple
This paper is motivated by interest in `non-associative non-commutative' probabilities over nonclassical propositional systems represented by orthomodular lattices. It is suggested that the appropriate structures of interest here will not be spaces with binary multiplication, but instead triple spaces. In particular the authors are concerned with tripotents (analogous to idempotents in binary systems), and claim to show here that the collection of tripotents in the complex Banach spaces known as JBW* algebras, when supplemented by a largest element, forms a complete lattice. In addition they purport to show that subsystems of this lattice which are associative, are equivalently described either by `norm-closed faces of the generalised normal state space' or by principal ideals in the JBW* triple. The paper is a technical exposition, and the results are not explicitly used to explore the wider issues of nonclassical propositional and probability structure which motivate it.
scientific article; zbMATH DE number 844062
scientific article