1. On Operator Norm Localization Property On Operator Norm Localization Property School of Mathematics School of Mathematics Fudan University Fudan University Xiaoman Chen & Xianjin Wan

2. Background

3. Background

4. Background

5. Background What is the operator norm localization property ? What is the operator norm localization property ? That is a local estimation property for us to estimate the norm of any operator in Roe algebra. That is a local estimation property for us to estimate the norm of any operator in Roe algebra.

6. Background The Box space : Let Γ be a finite generated residually finite group

7. Background Question: Is this mapping can be extended to the reduced Roe Algebras? In Gong-Wang-Yu’s paper “Geometrization of the Strong Novikov Conjecture of Residually finite groups”, they proved that

8. Background Application to K-theory

9. Definitions and Basic properties It is not difficult to prove that if Γ has finite asymptotic dimension, then the above lifting can be extended to the reduced Roe algebra. Generalize the finite asymptotic case, Guoliang Yu introduced the following definition

10. Definitions and Basic properties

19. Problems 1. What kinds of finite generated groups are being of operator norm localization property? 2. Do the operations of groups preserve operator norm localization property?

20. Key Propositions

29. Main Results

30. Idea of proof:

31. Main Results

32. Choose x=e

33. Main Results

34. Using the above Proposition and the infinite union theorem, we have

35. Main Results

36. Further Problems 1. Let Γ be a finite generated residually finite group with operator norm localization property.Are the reduced Roe algebra and maximal Roe algebra of its box space same? 2. Can we prove the Coarse Baum-Connes Conjecture in the case of operator norm localization property ?

37. Thank you ! Thank you !