# 臺灣博碩士論文加值系統

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 最近，Dawson和Feinstein證明了一個交換的Banach代數的拓樸穩定秩是一的話，那它的 Banach 代數整擴張的拓樸穩定秩也是一。在這篇論文中，我們提供了這個命題的部分相反結果：如果一個交換的C*-代數的某個Arens-Hoffman擴張的拓樸穩定秩是一的話，那這個C*-代數的拓樸穩定秩也是一。
 Recently, Dawson and Feinstein showed that a Banach algebra integral extension B of a commutative Banach algebra A of topological stable rank one is again of topological stable rank one. In this thesis, we provide a partial converse to this statement: If an Arens-Hoffman extension A® of a commutative C*-algebra A has topological stable rank one then A has topological stable rank one.
 Chapter 1: Introduction 1 Chapter 2: History and definition of dimensions 3 2.1 Historial remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Covering dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Topological stable ranks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter 3: Results 11 3.1 Notations and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
 [1] H. Choda, An Extremeal Property of the Polar Decomposition in von Neumann Algebras,Proc. Japan Acad., 46 (1970), 341-344.[2] G. Corach and F. D. Su arez, Thin Spectra and Stable Range Conditions, J. Funct. Anal.,81 (1988), 432-442.[3] T. W. Dawson and J. F. Feinstion, On the Denseness of the Invertible Group in BanachAlgebras, Proc. Am. Math. Soc., 131 (2003), no.9, 2831-2839.[4] D. Deckard and C. Pearcy, On Matrices Over the Ring of Continuous Complex ValuedFunctions on a Stonian Space, Proc. Am. Math. Soc., 14 (1963), no.2, 322-328.[5] D. Deckard and C. Pearcy, On Algebraic Closure in Function Algebras, Proc. Am.Math.Soc., 15 (1964), no.2, 259-263.[6] O. Hatori and T. Miura, On a Characterization of the Maximal Ideal Spaces of CommutativeC∗-Algebras in which Every Element is the Square of Another, Proc. Am. Math.Soc., 128 (1999), no.4, 1185-1189.[7] W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, 1948.[8] J. A. Lindberg, Integral Extensions of Commutative Banach Algebras, Can. J. Math., 25(1973), 673-686.[9] T. Miura and K. Niijima, On a Characterization of the Maximal Ideal Spaces of AlgebraicallyClosed Commutative C∗-Algebras, Proc. Am. Math. Soc., 131 (2003), no.9,2869-2876.[10] J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley, 1984.[11] J. R. Munkres, Topology, Prentice-Hall, Second Edition, 2000.[12] T. W. Palmer, Banach Algebras and General Theory of *-Algebras (vol. 1), Cambridge:CUP, 1994.[13] A. R. Pears, Dimension Theory of General Spaces, Cambridge:CUP, 1975.[14] N. T. Peck, Representation of Functions in C(X) by Means of Extreme Points, Proc.Am. Math. Soc., 18 (1967), no.1, 133-135.[15] M. A. Rieffel, Dimension and Stable Rank in K-theory of C∗-Algebras, Proc. Lond.Math. Soc., 46 (1983), no.3, 301-333.[16] G. Robertson, On the Density of the Invertible Group in C∗-Algebras, Proc. Edimb.Math. Soc., 20 (1976), 153-157.[17] O. Zariski and P. Samuel, Commutative algebra (vol. I), Van Nostrand, Princeton, 1968.
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 1 維度與整擴張

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