Study on Banach algebras from general and geometric topology view point

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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.