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Operators induced by certain hypercomplex systems

Alpay Daniel, Cho Ilwoo

Opuscula Mathematica

2023

Vol. 43, no. 3

275--333

In this paper, we consider a family {Ht}t∈R of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations {(C2, πt)} t∈R of the hypercomplex system {Ht}t∈R, and study the realizations πt(h) of hypercomplex numbers h ∈ Ht, as (2 × 2)-matrices acting on C2, for an arbitrarily fixed scale t ∈ R. Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.