Research on operating functions on function spaces

给出了半单交换Banach代数的最大正则子代数与Apostol代数（所有可分解乘法算子的集合）在极大理想空间中重合的一个充分条件.作为应用，我们研究了由具有自然谱的算子构成的一类Fourier乘子代数的结构。在具有典型操作函数的特殊情况下，我们对某些函数空间也证明了类似的现象。给出了交换C^*-代数到其子代数的弱投影存在的一个充分条件。研究了BKW算子在某些函数空间上的结构，包括圆盘代数上的BKW算子。在一定的条件下，刻画了紧区间上所有真实的值连续函数代数上的BKW算子，研究了Blaschke积的弱积，解决了Gorkin和Mortini关于素理想的一个问题.刻画了道格拉斯代数上的余维1等距.在交换C^*-代数的极大理想空间是局部连通的情况下，刻画了其中每个元素都是另一个元素的平方的极大理想空间.研究了交换Banach代数上的环同态，在空间情形下，利用极大理想空间上的映射和复数域上的环同态刻画了交换Banach代数上的环同态。作为结果的应用，我们证明了某些半单交换Banach代数上环同态的自动线性结果。特别地，我们证明了像包含非常数函数的圆盘代数上的环同态的线性性。

We gave a sufficient conditons for the greatest regular subalgebra and the Apostol algebra (the set of all decomposable multiplication operators) of semi-simple commutative Banach algebras coincide with each other in terms of the maximal ideal spaces. As an application, we investigated stuructures of subalgebras of certain Fourier multipliers which consists of operators with natural spectra. In a special case with typical operating functions, we showed similar phenomenun for certain function spaces. We gave a sufficient condition for the existence of weak projections from commutative C^*-algebras into its subalgebra. We investigated structures of BKW operators on certain function spaces including of the disk algebra. We also characterized BKW operators on the algebara of all real valued continuous functions on the compact intervals under certain additional conditions.We investigated weak products of Blaschke products, and solved a problem of Gorkin and Mortini on prime ideals. We characterized codimension 1 isometries on the Douglas algebras. We characterized the maximal ideal space of commutative C^*-algebra in which every element is the squareof another in case that the maximal ideal space is locally connected. We studied ring homomorphisms on commutative Banach algebras and in the spacial cases, we characterized in terms of mapping on the maximal ideal spaces and ring homomorphisms on the complex number field. As an application of the result we proved automatic linearity results for ring homomorphism on certain semi-simple Commutative Banach algebras. In particular, we proved linearity for ring homomorphisms on the disk algebras whose image contains non-constant functions.