Hypercomplex Representation of Finite-Dimensional Unital Archimedean f-Algebras

Abstract

In this paper, we characterize all N-dimensional hypercomplex numbers having unital Archimedean f-algebra structure. We use matrix representation of hypercomplex numbers to define an order structure on the matrix spectra. We prove that the unique (up to isomorphism) unital Archimedean f-algebra of hypercomplex numbers of dimension \(N \ge 1\) is that with real and simple spectrum. We also show that these number systems can be made into unital Banach lattice algebras and we establish some of their properties. Furthermore, we prove that every 2N-dimensional unital Archimedean f-algebra is the hyperbolization of that of dimension N. Finally, we consider hypercomplex number systems of dimension \(N=2,3,4,6\) and give their explicit matrix representation and eigenvalue operators. This work is a multidimensional generalization of the results obtained in Gargoubi and Kossentini (Adv Appl Clifford Algebras 26(4):1211–1233, 2016) and Bilgin and Ersoy S (Adv Appl Clifford Algebras 30:13, 2020) for, respectively, the two and four-dimensional systems.