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St. Petersburg Mathematical Journal

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


Authors: M. I. Belishev and A. F. Vakulenko
Translated by: M. I. Belishev
Original publication: Algebra i Analiz, tom 31 (2019), nomer 1.
Journal: St. Petersburg Math. J. 31 (2020), 1-12
MSC (2010): Primary 30F15; Secondary 35Q99, 46J99
DOI: https://doi.org/10.1090/spmj/1581
Published electronically: December 3, 2019
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Abstract: Let $ \mathscr {A}(D)$ be a Banach algebra of functions continuous in $ D=\{z\in \mathbb{C} \,\vert\, \vert z\vert\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} \,\vert\, 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 \,\vert\, \vert x\vert\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) \,\vert\, \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} \,\vert\, x_0\in B\}\cong B$, where $ \delta ^\mathbb{H}_{x_0}(p)=p(x_0)$.


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Additional Information

M. I. Belishev
Affiliation: St. Petersburg Branch, V. A. Steklov Mathematical Institute, Russian Academy of Sciences, St. Petersburg, Russia; St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg 199034, Russia
Email: belishev@pdmi.ras.ru, m.belishev@spbu.ru

A. F. Vakulenko
Affiliation: St. Petersburg Branch, V. A. Steklov Mathematical Institute, Russian Academy of Sciences, St. Petersburg, Russia
Email: vak@pdmi.ras.ru

DOI: https://doi.org/10.1090/spmj/1581
Keywords: 3d quaternion harmonic fields, real uniform Banach algebras, characters
Received by editor(s): March 26, 2018
Published electronically: December 3, 2019
Additional Notes: The first author was supported by RFBR grant 17-01-00529-a
The second author was supported by RFBR grant 18-01-00269
Dedicated: To the memory of Boris Sergeevich Pavlov
Article copyright: © Copyright 2019 American Mathematical Society