Operator Norm Localization Property

The index theory on the non-compact manifolds was developed by M.Atiyah and I.Singer in 1970's when they studied the index of the elliptic operators on the cover spaces.They found that the generalized elliptic operators are Fredholm operators relative to the the reduced group C*-algebras of the fundamental groups of manifolds and made progresses in some special cases,such as the index theorem on the foliated spaces obtained by A.Connes and G.Skandalis,the high index theorem on the homogenous spaces and their cover spaces by A.Connes and H.Moscovici,etc.In fact,the index of the generalized elliptic operators did not depend on the local geometry but the large scale structure of the manifolds.In 1980s,J.Roe studied the index theory of the complete Riemannian manifolds,enlighten by the index theory of the heat equations,he defined a class of C*-algebras C*(X) named Roe algebras by controlling the propagation of the locally compact operators.The Roe algebras reflect the large scale geometric structure,i.e.,the coarse structure of the spaces.Furthermore, the index of the generalized elliptic operators is an element of K-group of Roe algebras. So there is an assembly map from the coarse cohomology of space X to the K-group of its Roe algebra.The Baum-Connes assembly mapping provides a program to compute the K-theorem of Roe algebras.The coarse Baum-Connes conjecture states that the assembly map is an isomorphism,which implies the Novikov conjecture,Gromov-Lawson-Rosenber conjecture,idempotent conjecture of the group C*-algebras,etc.There are many advances on the coarse Baum-Connes conjecture in recent year. Most notably,Yu[78]has shown that the coarse Baum-Connes conjecture holds in case of X,which is coarsely embedded in Hilbert space.Using Gromov's expander graph structure Higson[45]gave a counterexample to the coarse Baum-Connes conjecture. The relevant construction is the box space X(Γ) of an infinite groupΓwith property T,residually finite and linear type,which is the coarse disjoint union of the quotient groupsΓ/Γ_n.Recently Gong,Wang and Yu[77]have established relations between the Coarse Geometric Novikov Conjecture for the box space X(Γ) and the Strong Novikov Conjecture for an infinite groupΓwith property T,residually finite. Since the Strong Novikov Conjecture holds for many infinite groups with property T, this implies that the Coarse Geometric Novikov Conjecture also holds for a large class of sequences of expanders.In Higson's original construction[44]and in Gong,Wang and Yu's construction[77],there is an algebraic lifting principle,that is,an operator T∈C*_alg(X(Γ)) will restrict an operator on C*_alg(Γ/Γ_n) for all but finitely many n, and such an operator can then be lifted to aΓ_n-invariant element of Roe algebra ofΓ.In general,such lifting can be extended to the maximal norm closure[77].Using some kind of localization estimation of operator norm in the case of asymptotic finite dimension,Higson[62]proved that the lifting can also be extended to the reduced norm closure.This was important in his original construction of counterexample to the Coarse Baum-Connes Conjecture.The natural question is what kind of coarse geometric conditions will be needed to guarantee the algebraic level lifting to extend to the reduced norm level.Guoliang Yu introduces the definition of operator norm localization property,which generalizes the local estimation property in the case of asymptotic finite dimension metric space.In this dissertation we use coarse geometric methods to study the permanence of the operator norm localization property.There are five chapters in our paper.The first chapter is the preface of this thesis.We introduce the operator norm localization property,some concepts of coarse geometry theory,index theory,and the latest developments.Furthermore,the problems discussed in this thesis and our main results will be introduced.In chapter two,we will discuss basic properties of the operator norm localization and metric sparsification property.We showed that both properties are coarse invariant. Some formulas of operator norm localization number and metric sparsification number are established.Furthermore,the relations between operator norm localization properties and metric sparsification properties are obtained.And in the last section of this chapter we give some nontrivial examples for operator norm localization and metric sparsification property.In chapter three,we will discuss the permanence of operator norm localization property of some operation on the metric spaces.We get that the operator norm localization property is preserved on the inductive limits and extensions of groups.And we show that the norm localization property is preserved by group action on a metric space which has metric sparsification property.Furthermore,we get the infinite union theory in some special case which we will need in the chapter four.In chapter four,we will discuss the invariance of operator norm localization property on the group operations.By using the infinite union theorem and permanence of groups acting on a metric spaces that we obtained in chapter four,we show that the op- erator norm localization property is invariant on the relatively hyperbolic groups,free products and amalgamated free products of groups,and fundamental groups of graph of groups.In chapter five,we prove that,if a metric space which has operator norm localization property,then its associated Roe algebra has no nontrivial ghost elements.