On the Scaled Hypercomplex Numbers: Quaternions through Split Quaternions
DOI:
https://doi.org/10.62298/advmath.7Keywords:
Scaled Hypercomplex Rings, Scaled Hypercomplex Monoids, Dynamical Systems, Free ProbabilityAbstract
In this paper, we analyze a certain algebraic structure ℋ[−1, 1] containing all t-scaled hypercomplex numbers of ℍt where the scales t are from −1 to 1, i.e., −1 ≤ t ≤ 1 in ℝ. The algebraic, operator-theoretic, and operator-algebraic properties of ℋ[−1, 1] are studied under the local dynamics on the closed interval [−1, 1] inherited from the dynamics on the continuum ℝ. Also, some analytic properties of an interesting type of operators switching scales of hypercomplex numbers acting on ℋ[−1, 1] are considered, and we investigate how they affect the analysis on ℋ[−1, 1].
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