Operators induced by certain hypercomplex systems

In this paper, we consider natural Hilbert-space representations ${\left\{({\mathbb{C}}^2,{\pi }_t)\right\}}_{t\in \mathbb{R}}$ of the hypercomplex system ${\left\{{\mathbb{H}}_t\right\}}_{t\in \mathbb{R}}$, and study the realizations ${\pi }_t\left(h\right)$ of hypercomplex numbers $h\in {\mathbb{H}}_t$, as $(2\times 2)$-matrices acting on ${\mathbb{C}}^2$, for an arbitrarily fixed scale $t\in \mathbb{R}$. Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.