Study on Banach algebras from general and geometric topology view point

本文的目的是利用一般拓扑学和几何拓扑学研究Banach代数的代数和几何。两个研讨会“巴拿赫代数研讨会”举行的支持下，目前的赠款（2006年。11. 15-16和2007年，11。21 - 22筑波大学。在研讨会讨论的基础上，对连续函数代数闭代数的刻画问题取得了一些进展。K. Kawamura，T. Miura，N. Brodskiy，J. Dydak和A. Karasev，连续函数的获得。K. Kawamura，T. Miura，N. Brodskiy，J. Dydak和A. Karasev等人得到了几个结果，证明了代数闭性强烈依赖于底层空间的第一可数性。T. Miura与D.本间还研究了函数代数上的其他类型的方程。O. Hatori，T. Miura与H。Oka，H.高木和S-E。Takahasi等研究了二元解析函数的半单单位Banach代数和一致代数上的保谱映射，得到了这类映射是同构或等距的一些条件。K. Sakai与S。Uehara，W. Kubis和K.我从无穷维拓扑流形的角度研究了一些超空间和函数空间。其中一个结果指出LF空间的开子集的拓扑类型是由它们的同伦类型决定的，这是LF流形理论的一个开拓性工作。H. Kato与E.松桥角Bingon研究了Bing映射和Krasinkewicz映射，这两个映射与光滑映射完全相反，但又是一般的，并且根据维数理论和拓扑动力学研究了遗传不可分解的连续统。K. Yamazaki研究了集值函数的延拓问题，并利用延拓性质给出了可数单调仿紧性的一个刻画。K. Eda和V. Matijevic研究了二维空间的覆盖同态。

The purpose of this research is to study algebra and geometry of Banach algebra by making use of general and geometric topology. Two workshops "Seminar on Banach algebras" were held by the support of the present grant (2006. 11. 15-16 & 2007, 11. 21 - 22 University of Tsukuba. On the basis of the discussion at the workshop, some progress was made on the characterization problem of algebraically closed algebras of continuous functions. K. Kawamura, with T. Miura, N. Brodskiy, J. Dydak and A. Karasev, obtained of continuous functions. K. Kawamura, with T. Miura, N. Brodskiy, J. Dydak and A. Karasev, obtained several results which demonstrate that the algebraic closedness strongly depends on the first-countability of the underlying space. T. Miura with D. Honma also investigate other types of equations on function algebras. O. Hatori, T. Miura with H. Oka, H. Takagi and S-E. Takahasi, investigated spectrum-preserving maps on semisimple unital Banach algebras and uniform algebras of analytic functions of two variables and obtained some conditions for these map to be isomorphisms or isometries. K. Sakai with S. Uehara, W. Kubis and K. Mine studied some hyperspaces and function spaces from the view point of infinite dimensional topological manifolds. One of the results states that the topological type of open subsets of LF space is determined by their homotopy types, a pioneering work on the theory of LF manifolds. H. Kato with E. Matsuhashi, C. Mouron investigated Bing maps and Krasinkewicz maps, the complete opposite to smooth maps yet generc, and hereditarily indecomposable continua in view of dimension theory and topological dynamics. K. Yamazaki studied extension problems of set-valued functions and gave a characterization of countable monotone paracompactness in terms of the extension property. K. Eda with V. Matijevic studied the covering homomorphism between two dimensional solenoids.