Research on algebraic equations with coefficients in Banach algebras

交换Banach代数的代数结构与其极大理想空间是相互联系的。研究Banach代数C（X）的代数元的谱对于研究其代数结构包括代数闭性具有重要意义。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。本文在不假定交换Banach代数之间的保谱映射是线性的情况下，证明了这类映射的若干结果。研究了COO的抽象n次根的存在性，证明了当X是局部连通的或第一可数的时，COO的n次根的存在性条件与代数闭性条件是等价的.本文给出了一个例子，当C（X）在第一次平方根闭的情况下，映射T既不是线性的，也不是乘法的，但对C（X）中的每一个元素f和g，TfTg的范数与fg的范数一致，最后给出了平方根闭的条件.我们研究了映射T，使得对任意的f和g，TfTg+1的范数与fg+1的范数一致，并证明了在某些情况下T是线性的和乘法的.这项研究是不完整的，但提供了新的和有趣的功能。本文证明了当局部紧交换群作用在冯诺依曼代数上时，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.