Spin factors as generalized Hermitian algebras
Let \(V\) be a real vector space with a symmetric positive definite bilinear form \(f:V\times V \to\mathbb R\). Positive-definite spin factor asociated with \((V,f)\) is the space \(R\oplus V\) equipped with the product \[ (\lambda+v)\circ (\mu+w)=\lambda\mu+f(v,w)+\lambda w+\mu v \] for all \(\lambda\), \(\mu\in\mathbb R\) and \(v,w\in V\). A Generalized Hermitian algebra (GH-algebra) is a subgroup \(G\) of the additive group of a ring \(R\) with the unit 1 such that \(G\) is a partially ordered abelian group with positive cone \(G^+\) satisfying certain requirements. A GH-algebra is of rank 2 if the projection structure of \(G\) is nontrivial and if every nontrivial projection is an atom. The paper analyzes the relationship between above mentioned structures. Main theorem states that positive-definite spin factors of dimension greater than one are mathematically equivalent to generalized Hermitian algebras of rank 2. For proving this result the authors derive many interesting properties of spin factors and GH-algebras.
scientific article; zbMATH DE number 5577215