On algebras of harmonic quaternion fields in ${\mathbb{R}}^3$

Let $\mathcal{A}(D)$ be an algebra of functions continuous in the disk $D=\{z\in \mathbb{C}\mathrm{\mid }\mathrm{\mid }z\mathrm{\mid }⩽1\}$ and {\it holomorphic} into $D$. The well-known fact is that the set $\mathcal{M}$ of its characters (homomorphisms $\mathcal{A}(D)\to \mathbb{C}$) is exhausted by the Dirac measures $\{{\delta }_{z_0}\mathrm{\mid }{z}_0\in D\}$ and a homeomorphism ${\mathscr M}\cong D$ holds. We present a 3d analog of this classical result as follows. Let $B=\{x\in {\mathbb{R}}^3\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }⩽1\}$. A quaternion field is a pair $p=\{\alpha ,u\}$ of a function $\alpha$ and vector field $u$ in the ball $B$. A field $p$ is {\it harmonic} if $\alpha ,u$ are continuous in $B$ and $\mathrm{\nabla }\alpha =\mathrm{r}\mathrm{o}\mathrm{t}u,\mathrm{d}\mathrm{i}\mathrm{v}u=0$ holds into $B$. The space $\mathcal{Q}(B)$ of such fields is not an algebra w.r.t. the relevant (point-wise quaternion) multiplication. However, it contains the commutative algebras ${\mathscr A}\omega(B)={p\in{\mathscr Q}(B),|,,\nabla\omega\alpha=0,,\nabla_\omega u=0},,(\omega\in S^2)$, each ${\mathcal{A}}_{\omega }(B)$ being isometrically isomorphic to ${\mathscr A}(D)$\mathrm{.}Thisenablesonetointroduceaset${\mathscr M}^{\mathbb H}$ofthe$\mathbb H$-valuedlinearfunctionalson${\mathscr Q}(B)({\it\mathbb H-characters}), which are multiplicative on each{\mathscr A}\omega(B)$, and prove that ${\mathscr M}^{\mathbb H}={\delta^{\mathbb H}{x_0},|,,x_0\in B}\cong B$,where$\delta^{\mathbb H}_{x_0}(p)=p(x_0)$.