一致Roe代数中的逼近性质
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摘要
粗的Roe代数和一致Roe代数是粗几何中两类非常重要的C*-代数,无论是在指标理论还是正合性问题都有着重要的作用.是联系几何、拓扑和分析的重要工具.本文主要研究一致Roe代数中的逼近性质,具体安排如下：
第一章是引言,我们通过回顾指标理论的一些结果介绍了我们所研究问题的背景,并且列出本文的主要研究结果.
第二章主要介绍与本文相关的基本知识,包括基本的粗几何知识,群的约化C*-代数以及度量空间和群的速降性质等等.
第三章分为两部分,第一部分研究了带有群作用的度量空间的一致Roe代数关于群作用不变的问题.设X是一个度量空间,群G等距的作用到X上.记Cu*(X)表示X的一致Roe代数,那么可以定义Cu*(X)中两个C*-子代数.一种方式是在Cu*(X)的有限传播算子中先取关于群作用不变的元素再取算子范数闭包,另一种方式就是直接在Cu*(X)中取群作用不变的算子.本文给出了一个这两个代数相等的充分条件.第二部分是非交换的Fejer定理.我们知道经典的Fourier分析中,单位圆周T上的连续函数的Fourier展开不一定一致收敛到这个函数本身.利用Fourier变换知道单位圆周上的连续函数的全体(记为C(T))与整数群的约化C*-代数(记为Cr*(Z))是同构的.换句话说,Cr*(Z)中的算子沿着对角线切割后得到的算子不一定依范数收敛到该算子本身.非交换的Fejer定理就上述这个结果进行了推广,即对于有限生成群的约化C*-代数和离散度量空间的一致Roe代数中的算子,它们沿着对角线切割后得到的算子不一定依范数收敛到该算子本身.文章还给出了这两个C*-代数中的算子沿着对角线切割得到的算子可以依范数收敛到该算子本身的充分条件.
第四章我们提出了一致Roe代数的等度nuclear性质,证明了一些nuclear的保持性问题.如：群G相对于子群G1,G2,…Gn是双曲群,那么Cu*(G)是nuclear的充要条件是Cu*(G1),Cu*(G2),…,Cu*(Gn)是具有nuclear性质.最后还说明了粗的Roe代数不具有nuclear性质.
The coarse Roe algebras and the uniform Roe algebras are two kinds of important C*-algebras and they play an important role in both index theory and exactness problem in C*-algebra theory. In this paper, we study some approximation properties in the uniform Roe algebras. The thesis is arranged as the following:
The first chapter is the introduction. We introduce the background of our problems and give our main results.
In chapter 2, we introduce some preliminaries which are used in the thesis. For example, the elementary knowledge of coarse geometry, the reduced C*-algebras, the rapid decay property for groups and the metric spaces and so on.
Chapter 3 contains two parts. In the first part, we studied the group invariant ap-proximation property of the uniform Roe algebra of the metric space with a group action. Let X be a metric space with a group G acting on X isometrically. Denote by Cu*(X) the uniform Roe algebra of X. We can define two C*-subalgebras of Cu*(X). One can be defined by taking the finite propagation operators in Cu*(X) which are in-variant under the group action first, then taking the hull of these operators. The other is defined by taking the group invariant operators in Cu*(X) directly. We will study when these two C*-algebras are identity. In the second part, we find the non-commutative Fejer theorem. Let T be the unit circle, C(T) be the set of all continuous functions on T. In the classical Fourier analysis, we know there are some elements in C(T) whose Fourier series are not converge to themselves uniformly. By Fourier transformation, C(T) is isomorphic to the reduced C*-algebras of the integer group Z. In other words, there are some operators in Cr*(Z) Which are not be approximated by the truncations along the diagonal of themselves. We extend this result to the finite generated groups and the discrete metric spaces. For the reduced C*-algebras of finite generated groups and the unform Roe algebras of the discrete metric spaces, there are operators which are not approximated by the truncations along the diagonal of themselves. Also, we give some sufficient condition for the operators in these two C*-algebras which can be approximated by the truncation along the diagonal of themselves.
In chapter 4, we give the definition of the equi-nuclearity of the uniform Roe alge-bras and prove some permanence of nuclearity. For example, let G be a group which is hyperbolic relative to the subgroups G1, G2,…, Gn, then Cu*(G) is nuclear if and only if Cu*(G1),Gu*(G2),…,Cu*{Gn) are nuclear. we also prove that the coarse Roe algebras are not nuclear.
第一章是引言,我们通过回顾指标理论的一些结果介绍了我们所研究问题的背景,并且列出本文的主要研究结果.
第二章主要介绍与本文相关的基本知识,包括基本的粗几何知识,群的约化C*-代数以及度量空间和群的速降性质等等.
第三章分为两部分,第一部分研究了带有群作用的度量空间的一致Roe代数关于群作用不变的问题.设X是一个度量空间,群G等距的作用到X上.记Cu*(X)表示X的一致Roe代数,那么可以定义Cu*(X)中两个C*-子代数.一种方式是在Cu*(X)的有限传播算子中先取关于群作用不变的元素再取算子范数闭包,另一种方式就是直接在Cu*(X)中取群作用不变的算子.本文给出了一个这两个代数相等的充分条件.第二部分是非交换的Fejer定理.我们知道经典的Fourier分析中,单位圆周T上的连续函数的Fourier展开不一定一致收敛到这个函数本身.利用Fourier变换知道单位圆周上的连续函数的全体(记为C(T))与整数群的约化C*-代数(记为Cr*(Z))是同构的.换句话说,Cr*(Z)中的算子沿着对角线切割后得到的算子不一定依范数收敛到该算子本身.非交换的Fejer定理就上述这个结果进行了推广,即对于有限生成群的约化C*-代数和离散度量空间的一致Roe代数中的算子,它们沿着对角线切割后得到的算子不一定依范数收敛到该算子本身.文章还给出了这两个C*-代数中的算子沿着对角线切割得到的算子可以依范数收敛到该算子本身的充分条件.
第四章我们提出了一致Roe代数的等度nuclear性质,证明了一些nuclear的保持性问题.如：群G相对于子群G1,G2,…Gn是双曲群,那么Cu*(G)是nuclear的充要条件是Cu*(G1),Cu*(G2),…,Cu*(Gn)是具有nuclear性质.最后还说明了粗的Roe代数不具有nuclear性质.
The coarse Roe algebras and the uniform Roe algebras are two kinds of important C*-algebras and they play an important role in both index theory and exactness problem in C*-algebra theory. In this paper, we study some approximation properties in the uniform Roe algebras. The thesis is arranged as the following:
The first chapter is the introduction. We introduce the background of our problems and give our main results.
In chapter 2, we introduce some preliminaries which are used in the thesis. For example, the elementary knowledge of coarse geometry, the reduced C*-algebras, the rapid decay property for groups and the metric spaces and so on.
Chapter 3 contains two parts. In the first part, we studied the group invariant ap-proximation property of the uniform Roe algebra of the metric space with a group action. Let X be a metric space with a group G acting on X isometrically. Denote by Cu*(X) the uniform Roe algebra of X. We can define two C*-subalgebras of Cu*(X). One can be defined by taking the finite propagation operators in Cu*(X) which are in-variant under the group action first, then taking the hull of these operators. The other is defined by taking the group invariant operators in Cu*(X) directly. We will study when these two C*-algebras are identity. In the second part, we find the non-commutative Fejer theorem. Let T be the unit circle, C(T) be the set of all continuous functions on T. In the classical Fourier analysis, we know there are some elements in C(T) whose Fourier series are not converge to themselves uniformly. By Fourier transformation, C(T) is isomorphic to the reduced C*-algebras of the integer group Z. In other words, there are some operators in Cr*(Z) Which are not be approximated by the truncations along the diagonal of themselves. We extend this result to the finite generated groups and the discrete metric spaces. For the reduced C*-algebras of finite generated groups and the unform Roe algebras of the discrete metric spaces, there are operators which are not approximated by the truncations along the diagonal of themselves. Also, we give some sufficient condition for the operators in these two C*-algebras which can be approximated by the truncation along the diagonal of themselves.
In chapter 4, we give the definition of the equi-nuclearity of the uniform Roe alge-bras and prove some permanence of nuclearity. For example, let G be a group which is hyperbolic relative to the subgroups G1, G2,…, Gn, then Cu*(G) is nuclear if and only if Cu*(G1),Gu*(G2),…,Cu*{Gn) are nuclear. we also prove that the coarse Roe algebras are not nuclear.
引文
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[15]X. Chen, M. Dadarlat, E. Guentner and G. Yu, Uniform embedability and exactness of free products, J. Funct. Anal.,205 (2003),293-314.
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[18]X. Chen and Q. Wang, Notes on Ideals of Roe Algebras, Oxford Quarterly J. of Math.,52 (2001),437-446.
[19]X. Chen and Q. Wang, Ideal structure of uniform Roe algebras of coarse spaces,J. Funct. Anal.216 (2004), no.1,191-121.
[20]X. Chen and Q. Wang, Ghost ideals in uniform Roe algebras of coarse spaces, Arch. Math. (Basel),84 (2005), no.6,519-526.
[21]X. Chen and Q. Wang, Asymptotic coarsening of proper metric spaces and index problems, Manuscriota Math.,110 (2003),475-486.
[22]X. Chen and X. Wang, Operator norm localization property of relative hyperbolic groups and graphy of groups, J. Funct. Anal.255 (2008),642-652(eletronic).
[23]X. Chen and S. Wei, Spectral invariant subalgebras of reduced crossed product C*-algebras, J. Funct. Anal.197 (2003),228-246.
[24]P. A. Cherix, M. Cowling, P. Jolissaint, P. Julg and A. Valette, Groups with the Haagerup property (Gromov's a-T-amenability), Birkhauser, Boston,2001.
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[27]M. Cowling and U. Haagerup, Completely bounded multiplier of the Fourier algebra of a simple Lie group of real ranl one, Invent. Math.,96 (1989),507-549.
[28]M. Dadarlat and E.Guentner, Constructions preserving Hilbert space uniform embeddability of discrete grouops, Vol 335, Trans. Amer. Math. Soc. No.8 (2003),3253-3257.
[29]M. Dadarlat and E. Guentner, Uniform embeddabilty of relatively hyperbolic groups, preprint 2006.
[30]C. Drutu and M. Sapir, Relatively hyperbolic groups with rapid decay property, IMRN,19 (2005),1182-1194.
[31]A. Dranishnikov, On Generalized amenability, preprint.
[32]R. Edwards, Fourier Series, A Modern Introduction, Volume 1, Graduate Texts in Mathemat-ics 64, Springer-Verlag.
[33]B. Farb, Relatively hyperbolic groups, GAFA,8 (1998),810-840.
[34]G. Fendler, Weak amenability of Coxeter groups, arxiv:math/0203052vl,2002.
[35]S. Ferry, A. Ranicki and J. Rosenberg (eds.), Novikov conjectures index theorems and rigid-ity,Londond Math. Soc. Lecture Notes, nos.226,227, Cambridge University Press,1995.
[36]G. Gong, Q. Wang and G. Yu, Geometrization of the Strong Novikov Conjecture for residually finite groups, J. Reine Ang. Math.621 (2008),159-189.
[37]F. Greenleaf, Invariant means on topological groups, New York University,1969.
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[39]M. Gromov, Asymptotic invariants of infinite groups,London Math. Soc., Lecture Notes Series 182, Geometric groups theory, Vol.2, CUP 1993.
[40]M. Gromov, Spaces and questions, GAFA 2000, Special Volume, Part Ⅰ, Birkhddota Verlag, Basel(2000),118-161.
[41]E. Guentner, Exactness of the one relator groups, Proc. Amer. Math. Soc.130 (2002), no.4, 1087-1093 (eletronic).
[42]E. Guentner, N. Higson Weak amenability of CAT(0)-cubical groups, arxiv:math. OA/0702568V 1,2007.
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