论文标题
嵌入式riemannian歧管上的纺纱场的Gelfand变换
A Gelfand Transform for Spinor Fields on Embedded Riemannian Manifolds
论文作者
论文摘要
Gelfand的经典结果表明，交换Banach代数上拓扑的字符谱对基础空间是同质的。该事实用于通过边界控制（BC）方法在维度2中解决Calderón问题。为了在维度3中应用BC方法，可以用谐波四元素场的空间代替复杂的全态函数的代数，但是该空间不再是代数，也不是交换性的。尽管如此，已经显示出存在频谱的合适概念，而在基础空间为凸的情况下，频谱对球是同构的。我们的目标是将此结果概括为任意维度的更一般的歧管。为此，我们使用多骑场的Clifford代数，我们关心的一组功能是空间单基因旋场。该光谱由旋转型功能组成，这些功能尊重单基因纺纱场空间的模块和亚代词结构。我们证明，对于$ \ mathbb {r}^n $中的紧凑型区域，该频谱对歧管本身是同质的。此外，一些基本的引理和命题被证明是任意紧凑的riemannian $ n $ -manifolds。最后，我们以一块石头 - 韦尔斯特拉斯定理结尾，显示了由单基因旋转磁场在任意紧凑型$ m $上产生的代数，边界在连续旋转磁场的代数中密集。
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 Calderón 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.
