Research on Fourier multiplier by operating functions on function spaces

本文研究了局部紧阿贝尔群g上的傅里叶乘子，得到了一个紧阿贝尔群上p-q乘子的极大理想空间。特别证明了G的对偶群在极大理想空间上是稠密的，从而证明了p-q乘法器谱的力的自然性。确定了p-2乘法器的工作函数。设C_0M_p (G)表示L^p乘子的代数它的傅里叶变换在无穷远处消失。证明了Apostol代数与最大正则闭子代数RegC_0M_p (G)重合，在某种意义上，它们在C_0M_p (G)上是极大的。这个证明依赖于用傅里叶乘数建模的连续函数的抽象代数的一般结果。同时证明了如果代数的极大理想空间中，它们的最大正则闭子代数与具有自然谱的函数集重合。Laursen和Neumann证明了如果p=1或G是紧的，则RegC_0M_p (G)是G对偶群中由其Gelfand变换消失的乘子组成的闭理想C_<00>M_p (G)，证明了如果p*1，则RegC_0M_p （R^n）不是C_0M_p （R^n）和C_<00>M_p (R^n)={0}的理想。设G是一个非离散的局部紧阿贝尔群。证明了在具有自然谱的L^1 (G)的根外存在有界正则Borel测度。特别是当G不紧时，则测度的Fourier-Stieltjes变换可以在对偶群上的无穷远处消失，从而回答了Eschimier、Laursen和Neumann提出的问题。我也学习bse代数。

In this research I study Fourier multiplier on locally compact abelian groups G.The maximal ideal space of p-q multplier on a comact abelian group is identified. In particular, I proved that the dual group of G is dense in the maximal ideal space, henceforce naturality of sspectra of p-q multiplier is proved. The operating functions of p-2 multplier is also identified. Let C_0M_p (G) denote the algebra of L^p-multiplier whose Fourier transforms vanish at infinity. I proved that the Apostol algebra coincides with the greatest regular closed subalgebra RegC_0M_p (G) and they are maximal, in a sense, in C_0M_p (G). The proof depends on the general results concerning abstract algebras of continuous functions which are modeled after Fourier multipliers. I also proved that if the maximal ideal space of the algebra in thin, they the greatest regular closed subalgebra coincides with the set of functions with natural spectra. Laursen and Neumann proved that if p=1 or G is compact, then RegC_0M_p (G) is the closed ideal C_<00>M_p (G) which consists of multplier whose Gelfand transforms vanish of the dual group of G.I proved that if p*1, then RegC_0M_p (R^n) is not an ideal of C_0M_p (R^n) and C_<00>M_p (R^n)= {0}. Let G be a non-discrete locally comapct abelian group. I prove that there exists a bounded regular Borel measure outside of the radical of L^1 (G) with a natural spectrum. In particular if G is not compact, then the Fourier-Stieltjes transform of the measure can be vanish at infinity on the dual group, which answers the question posed by Eschimier, Laursen and Neumann. I also study BSE-algebras.