Research on Fourier multiplier by operating functions on function spaces

本文研究了局部紧交换群G上的Fourier乘子，给出了紧交换群G上p-q乘子的极大理想空间。特别地，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s谱的自然性。并对p-2乘法器的运算功能进行了分析。设C_0M_p（G）表示Fourier变换在无穷远处为零的L^p-乘子的代数。证明了Apostol代数与最大正则闭子代数RegC_0M_p（G）重合，并且它们在某种意义上都是C_0M_p（G）中的极大子代数。证明依赖于抽象代数的连续函数的傅立叶乘子后建模的一般结果。证明了若该代数的极大理想空间是薄的，则其极大正则闭子代数与具有自然谱的函数集一致。Laursen和Neumann证明了如果p=1或G是紧的，则RegC_0M_p（G）是<00>由G的对偶群的Gelfand变换为零的乘子构成的闭理想C_ M_p（G）.我证明了如果p*1，则RegC_0M_p（R^n）不是C_0M_p（R ^n）的理想，C_<00>M_p（R^n）= {0}.设G是非离散的局部紧交换群。证明了L^1（G）的根外存在一个有界正则Borel测度，且该测度具有自然谱.特别地，如果G不是紧的，则测度的Fourier-Stieltjes变换在对偶群上无穷远处为零，这回答了Schumimier，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.