Study on Banach algebras from general and geometric topology view point

本研究的目的是利用一般拓扑学和几何拓扑学来研究Banach代数的代数和几何。在这笔赠款(2006年)的支持下，举办了两次“Banach代数研讨会”。2007年11月15日至16日和2007年11月21日至22日筑波大学。在研讨会讨论的基础上，对连续函数的代数闭代数的刻画问题取得了一些进展。K.Kawamura，T.Miura，N.Brodski，J.Dydak和A.Karasev，得到了连续函数。K.Kawamura和T.Miura，N.Brodski，J.Dydak和A.Karasev得到了几个结果，证明了代数闭性强烈地依赖于基础空间的第一可数。T.Miura和D.Honma还研究了函数代数上的其他类型的方程。Hatori，T.Miura与H.Oka，H.Takagi和S-E。Takahasi，研究了半单单位Banach代数和二元解析函数一致代数上的保谱映射，得到了这些映射是同构或同构的一些条件。K.Sakai和S.Uehara，W.Kubis和K.mine从无限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空间开子集的拓扑型由其同伦类型决定，这是LF流形理论的开创性工作之一。H.Kato和E.Matsuhashi，C.Mouron从维度论和拓扑动力学的角度研究了Bing映射和Krasinkewicz映射，它们是光滑映射的完全相反的一般形式，以及遗传不可分解的连续体。K.Yamazaki研究了集值函数的延拓问题，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研究了二维螺线管之间的覆盖同态。

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.