In this paper we apply the method of functional identities to the study of group gradings by an abelian group ▫$G$▫ on simple Jordan algebras, under very mild restrictions on the grading group or the base field of coefficients.
COBISS.SI-ID: 15803225
The paper begins with short proofs of classical theorems by Frobenius and (resp.) Zorn on associative and (resp.) alternative real division algebras. These theorems characterize the first three (resp. four) Cayley-Dickson algebras. Then we introduce and study the class of real unital nonassociative algebras in which the subalgebra generated by any nonscalar element is isomorphic to ▫$\mathbb{C}$▫. We call them locally complex algebras. In particular, we describe all such algebras that have dimension at most 4. Our main motivation, however, for introducing locally complex algebras is that this concept makes it possible for us to extend Frobenius' and Zorn's theorems in a way that it also involves the fifth Cayley-Dickson algebra, the sedenions.
COBISS.SI-ID: 15758681
We characterize weak▫$^\ast$▫ closed unital vector spaces of operators on a Hilbert space ▫$H$▫. More precisely, we first show that an operator system, which is the dual of an operator space, can be represented completely isometrically and weak▫$^\ast$▫ homeomorphically as a weak▫$^\ast$▫ closed operator subsystem of ▫$B(H)$▫. An analogous result is proved for unital operator spaces. Finally, we give some somewhat surprising examples of dual unital operator spaces.
COBISS.SI-ID: 15862617