Research on algebraic equations with coefficients in Banach algebras

交换巴拿赫代数的代数结构及其极大理想空间是相互关联的。研究代数元的谱对于研究给定紧致Hausdorff空间X上所有复值连续函数的Banach代数C(X)的代数闭性等代数结构具有重要意义，因此研究两个Banach代数之间的保谱映射对于研究C(X)的代数结构具有重要意义。在不假设可交换巴拿赫代数上的线性的情况下，证明了保谱映射的几个结果。我们研究了Karahanjan引入的抽象意义上的COO的n次根的存在性，并证明了当X是局部连通或第一可数时，该条件等价于代数闭性。在C(X)第一次为平方根封闭的情况下，我们给出了一个映射T既非线性也非乘法的例子，而TfTg的范数与C(X)中每个元素f和g的范数重合，最后减少了平方根封闭的条件。我们研究了映射T，使得TfTg+1的范数与fg+1的范数对每一个f和g重合，并证明了T在某些情况下是线性的和乘法的。本研究虽不完整，但具有新颖有趣的特点。当局部紧阿贝尔群作用于von Neumann代数时，证明了Arveson谱子空间的Takesaki对偶定理。

Algebraic structure of commutative Banach algebra and thier maximal ideal spaces are related each other. To investigate the spectrum of the algebra element is important for the study of algebraic structure including of algebraically closedness of the Banach algebra C(X) of all complex-valued continuous functions on a given compact Hausdorff space X. Thus it is important to study spectrum-preserving maps between two Banach algebras for the research on algebraic structure of C(X). We have proved several results on spectrum-preserving maps between commutative Banach algebras without assuming linearity on the maps. We investigate the existence of n-th root in the abstract sense, which was introduced by Karahanjan, for COO, and we proved the condition is equivalent for algebraically closedness if X is locally connected or first countable. We show an example of the map T which is not linear nor multiplicative while the norm of TfTg coincides with the norm of fg for every element f and g in C(X) for the case where C(X) is square-root closed at the first time, and Finally reduce the condition on the square-root closedness. We study the maps T such that the norm of TfTg+1 coincides with the norm of fg+1 for every f and g and show that T is linear and multiplicative in certain cases. This research is not complete but give new and interesting feature. We prove a Takesaki duality theorem for Arveson's spectral subspaces if a locally compact abelian group acts on the von Neumann algebra.