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

Abstract

Let ${\mathscr A}(D)$ be an algebra of functions continuous in the disk $D=\{z\in{\mathbb C}\,|\,\,\,|z|\leqslant 1\}$ and {\it holomorphic} into $D$. The well-known fact is that the set ${\mathscr M}$ of its characters (homomorphisms ${\mathscr A}(D)\to\mathbb C$) is exhausted by the Dirac measures $\{\delta_{z_0}\,|\,\,z_0\in D\}$ and a homeomorphism ${\mathscr M}\cong D$ holds. We present a 3d analog of this classical result as follows. Let $B=\{x\in{\mathbb R}^3\,|\,\,|x|\leqslant 1\}$. 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 $\nabla\alpha={\rm rot\,}u,\,{\rm div\,}u=0$ holds into $B$. The space ${\mathscr Q}(B)$ 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 ${\mathscr A}_\omega(B)$ being isometrically isomorphic to ${\mathscr A}(D)$. This enables one to introduce a set ${\mathscr M}^{\mathbb H}$ of the $\mathbb H$-valued linear functionals on ${\mathscr Q}(B)$ ({\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 $\delta^{\mathbb H}_{x_0}(p)=p(x_0)$.