The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon thei...

Motivated by an Arens regularity problem, we introduce the concepts of matrix Banach space and matrix Banach algebra. The notion of matrix normed space in the sense of Ruan is a special case of our matrix normed system. A matrix Banach algebra is a matrix Banach space with a completely contractive multiplication. We study the structure of matrix Banach spaces and matrix Banach algebras. Then we...

The notion of strongly Lie zero-product preserving maps on normed algebras as a generalization of Lie zero-product preserving maps are dened. We give a necessary and sufficient condition from which a linear map between normed algebras to be strongly Lie zero-product preserving. Also some hereditary properties of strongly Lie zero-product preserving maps are presented. Finally the second dual of...

motivated by an arens regularity problem, we introduce the concepts of matrix banach space and matrix banach algebra. the notion of matrix normed space in the sense of ruan is a special case of our matrix normed system. a matrix banach algebra is a matrix banach space with a completely contractive multiplication. we study the structure of matrix banach spaces and matrix banach algebras. then we...

In this essay, we immerse into the framework of normed division algebras as a suitable arena to accommodate standard model elementary particles, and explore some applications cosmology. Remarkably, they permit interesting non-trivial realisations cosmological principle with an interplay between symmetry groups quaternions octonions. We also argue how these give rise potentially observational si...

Recent studies reveal that branching bisimilarity is decidable for both nBPP (normed Basic Parallel Processes) and nBPA (normed Basic Process Algebras). These results lead to the question if there are any other models in the hierarchy of PRS (Process Rewrite Systems) whose branching bisimilarity is decidable. It is shown in this paper that the branching bisimilarity for both nOCN (normed One Co...

Abstract We provide a proof of Jacobson’s theorem on derivations primitive rings with nonzero socle. Both and its formulation (in terms the so-called differential operators left vector spaces over division ring) underlie our paper. apply to describe standard operator real, complex, or quaternionic normed spaces. Indeed, when space is infinite-dimensional, every derivation such ring form $$A\rig...