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On Operator Norm Localization Property

This research explores the operator norm localization property as a local estimation technique for estimating the norm of operators in Roe algebras. Specifically, we analyze its implications for finite generated residually finite groups, following the work of Gong-Wang-Yu on the Strong Novikov Conjecture. We also consider the potential extension of mappings to reduced Roe algebras and provide definitions and basic properties necessary for understanding this framework. Additionally, we discuss related challenges, such as the relationship between operator norm localization and various types of groups, and pose further questions about the Coarse Baum-Connes Conjecture.

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On Operator Norm Localization Property

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

  2. Background

  3. Background

  4. Background

  5. Background 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.

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

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

  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

  11. Definitions and Basic properties

  12. Definitions and Basic properties

  13. Definitions and Basic properties

  14. Definitions and Basic properties

  15. Definitions and Basic properties

  16. Definitions and Basic properties

  17. Definitions and Basic properties

  18. Definitions and Basic properties

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

  20. Key Propositions

  21. Key Propositions

  22. Key Propositions

  23. Key Propositions

  24. Key Propositions

  25. Key Propositions

  26. Key Propositions

  27. Key Propositions

  28. Key Propositions

  29. Main Results

  30. Main Results Idea of proof:

  31. Main Results

  32. Main Results Choose x=e

  33. Main Results

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

  35. Main Results

  36. Further Problems • 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? • Can we prove the Coarse Baum-Connes Conjecture in the case of operator norm localization property ?

  37. Thank you !

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