A Gelfand Transform for Spinor Fields on Embedded Riemannian Manifolds

TL;DR

This paper generalizes the Gelfand transform to spinor fields on embedded Riemannian manifolds, establishing a spectral theory that links algebraic structures of monogenic spinor fields to the geometry of the manifold.

Contribution

It introduces a new spectral framework for monogenic spinor fields on general manifolds using Clifford algebras, extending previous results beyond convex domains.

Findings

Spectrum of monogenic spinor fields homeomorphic to the manifold

Spectral characterization applies to arbitrary compact Riemannian manifolds

Algebra generated by spinor fields is dense in continuous spinor fields

Abstract

A classical result of Gelfand shows that the topologized spectrum of characters on commutative Banach algebra is homeomorphic to the underlying space. This fact is used in solving the Calder\'on problem in dimension 2 via the boundary control (BC) method. To apply the BC method in dimension 3, the algebra of complex holomorphic functions can be replaced by the space of harmonic quaternion fields, but this space is no longer an algebra and is not commutative. Nonetheless, it has been shown that a suitable notion of a spectrum exists and in the case when the underlying space is convex, the spectrum is homeomorphic to the ball. Our goal is to generalize this result to more general manifolds in arbitrary dimension. To do so, we use Clifford algebras of multivector fields and the set of functions we care about is the space monogenic spinor fields. The spectrum consists of spinor valued functionals that respect the module and subalgebra structure of the space of monogenic spinor fields. We prove that for compact regions in $\mathbb{R}^n$, the spectrum is homeomorphic to the manifold itself. Furthermore, some essential lemmas and propositions are proven for arbitrary compact Riemannian $n$-manifolds. Finally, we end with a Stone--Weierstrass theorem which shows the algebra generated by the monogenic spinor fields on arbitrary compact $M$ with boundary is dense in the algebra of continuous spinor fields.