Research on Fourier multiplier by operating functions on function spaces

本文研究了局部紧交换群G上的傅立叶乘子，得到了紧交换群上p-q乘子的最大理想空间。特别地，我证明了G的对偶群在极大理想空间中是稠密的，从而证明了p-q乘子谱的强制自然性。确定了P-2倍增器的工作功能。设C_0M_p(G)表示其傅里叶变换在无穷远处消失的L p乘子的代数。证明了Apostol代数与最大正则闭子代数RegC0Mp(G)重合，在某种意义上，它们在C0Mp(G)中是极大的。证明依赖于关于连续函数抽象代数的一般结果，这些抽象代数是以傅立叶乘子为模型的。证明了：如果代数的极大理想空间是薄的，则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)是由G的对偶群的Gelfand变换为零的乘子组成的闭理想C_0M_p(G)。I证明了如果p*1，则RegC_0M_p(R^n)不是C_0M_p(R^n)和C_0M_p(R^n)={0}的理想。设G是非离散局部可交换群。证明了在L^1(G)的根外存在具有自然谱的有界正则Borel测度。特别地，如果G不是紧的，则测度的Fourier-Stieltjes变换可以在对偶群上无穷远为零，这回答了Eschimier、Laursen和Neumann提出的问题。我也研究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.