Free-Product Banach Algebras Induced by Scaled Hypercomplex Numbers

In this paper, we construct certain Banach algebras \(\left\{ \mathscr {M}_{t}\right\} _{t\in \mathbb {R}}\) ${\left\{{\mathcal{M}}_t\right\}}_{t\in \mathbb{R}}$ (over the complex field \(\mathbb {C}\) $\mathbb{C}$ ) generated by the multiplicative algebraic structure \(\left\{ \mathcal {G}_{t}\right\} _{t\in \mathbb {R}}\) ${\left\{{\mathcal{G}}_t\right\}}_{t\in \mathbb{R}}$ , embedded in the \(\mathbb {R}\) $\mathbb{R}$ -algebras \(\left\{ \mathbb {H}_{t}\right\} _{t\in \mathbb {R}}\) ${\left\{{\mathbb{H}}_t\right\}}_{t\in \mathbb{R}}$ of the scaled hypercomplex numbers. We study free probability on \(\left\{ \mathscr {M}_{t}\right\} _{t\in \mathbb {R}}\) ${\left\{{\mathcal{M}}_t\right\}}_{t\in \mathbb{R}}$ under bounded linear functionals \(\left\{ \tau _{t}\right\} _{t\in \mathbb {R}}\) ${\left\{{\tau }_t\right\}}_{t\in \mathbb{R}}$ , respectively, i.e., the free-probabilistic information on the Banach probability space \(\left\{ \left( \mathscr {M}_{t},\tau _{t}\right) \right\} _{t\in \mathbb {R}}\) ${\left\{\left({\mathcal{M}}_t,,,{\tau }_t\right)\right\}}_{t\in \mathbb{R}}$ are considered. And then, we construct the free-product Banach algebra \(\mathscr {M}_{t_{1},...,t_{N}}=\overset{N}{\underset{l=1}{\star }}\mathscr {M}_{t_{l}}\) ${\mathcal{M}}_{t_1,...,t_N}=\stackrel{N}{\underset{l=1}{\star }}{\mathcal{M}}_{t_l}$ for arbitrarily fixed multi scales \(t_{1},...,t_{N}\in \mathbb {R}\) $t_1,...,t_N\in \mathbb{R}$ , for any \(N\in \mathbb {N}\setminus \left\{ 1\right\} \) $N\in \mathbb{N}\backslash \left\{1\right\}$ , and study the free probability on \(\mathscr {M}_{t_{1},...,t_{N}}\) ${\mathcal{M}}_{t_1,...,t_N}$ . Our free-distributional data preserves certain analytic data on the scaled hypercomplex numbers \(\overset{N}{\underset{l=1}{\cup }}\mathbb {H}_{t_{l}}\) $\stackrel{N}{\underset{l=1}{\cup }}{\mathbb{H}}_{t_l}$ .