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

Abstract:

Let $\mathscr {A}(D)$ be a Banach algebra of functions continuous in $D=\{z\in \mathbb {C} | |z|\leq 1\}$ and holomorphic inside $D$. It is well known that the set $\mathscr {M}^\mathbb C$ of its characters (homomorphisms $\mathscr {A}(D)\to \mathbb {C}$) is exhausted by the Dirac measures $\{\delta _{z_0} | z_0\in D\}$ and we have homeomorphism $\mathscr {M}^\mathbb {C}\cong D$. The following 3d analog of this classical result is provided.

Let $B=\{x\in \mathbb {R}^3 | |x|\leq 1\}$. Quaternion fields are the pairs $p=\{\alpha ,u\}$, where $\alpha$ is a function and $u$ is a vector field in $B$, with the pointwise multiplication $pp’=\{\alpha \alpha ’-u\cdot u’, \alpha u’+\alpha ’u+u\wedge u’\}$. A field $p$ is harmonic if $\alpha , u$ are continuous in $B$ and $\nabla \alpha =\mathrm {curl}\thinspace u, \mathrm {div}\thinspace u=0$ inside $B$. The space $\mathscr {Q}(B)$ of harmonic fields is not an algebra but contains the subspaces-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 $\mathscr {A}(D)$. Let $\mathscr {M}^\mathbb {H}$ be the set of $\mathbb {H}$-valued linear functionals on $\mathscr {Q}(B)$ ($\mathbb {H}$-characters) that are multiplicative on each $\mathscr {A}_\omega (B)$. It is shown that $\mathscr {M}^\mathbb {H}=\{\delta ^\mathbb {H}_{x_0} | x_0\in B\}\cong B$, where $\delta ^\mathbb {H}_{x_0}(p)=p(x_0)$.