On Operator Norm LocalOn Operator Norm Localization Propertyization Property

School of Mathematics School of Mathematics Fudan UniversityFudan UniversityXiaoman Chen & Xianjin WanXiaoman Chen & Xianjin Wan

BackgroundBackground

BackgroundBackground

BackgroundBackground

BackgroundBackground

What is the operator norm localizatioWhat is the operator norm localization property ?n property ?

That is a local estimation property fThat is a local estimation property for us to estimate the norm of any operator us to estimate the norm of any operator in Roe algebra.or in Roe algebra.

BackgroundBackgroundThe Box space : Let The Box space : Let ΓΓ be a finite generated be a finite generated

residually finite groupresidually finite group

BackgroundBackground

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

BackgroundBackground

Application to K-theoryApplication to K-theory

Definitions and Basic propertiesDefinitions 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

Definitions and Basic propertiesDefinitions and Basic properties

Definitions and Basic propertiesDefinitions and Basic properties

Definitions and Basic propertiesDefinitions and Basic properties

Definitions and Basic propertiesDefinitions and Basic properties

Definitions and Basic propertiesDefinitions and Basic properties

Definitions and Basic propertiesDefinitions and Basic properties

Definitions and Basic propertiesDefinitions and Basic properties

Definitions and Basic propertiesDefinitions and Basic properties

Definitions and Basic propertiesDefinitions and Basic properties

ProblemsProblems

1.1. What kinds of finite generated What kinds of finite generated groups are being of operator norm groups are being of operator norm localization property?localization property?

2.2. Do the operations of groups preserve Do the operations of groups preserve operator norm localization property? operator norm localization property?

Key PropositionsKey Propositions

Key PropositionsKey Propositions

Key PropositionsKey Propositions

Key PropositionsKey Propositions

Key PropositionsKey Propositions

Key PropositionsKey Propositions

Key PropositionsKey Propositions

Key PropositionsKey Propositions

Key PropositionsKey Propositions

Main ResultsMain Results

Main ResultsMain ResultsIdea of proof:

Main ResultsMain Results

Main ResultsMain Results

Choose x=e

Main ResultsMain Results

Main ResultsMain ResultsUsing the above Proposition and the infinite union theorem, we have

Main ResultsMain Results

Further ProblemsFurther Problems1.1. Let Let ΓΓ be a finite generated residually finite grou be a finite generated residually finite grou

p with operator norm localization property .Are tp with operator norm localization property .Are the reduced Roe algebra and maximal Roe algebrhe reduced Roe algebra and maximal Roe algebra of its box space same?a of its box space same?

2.2. Can we prove the Coarse Baum-Connes ConjectCan we prove the Coarse Baum-Connes Conjecture in the case of operator norm localization proure in the case of operator norm localization property ?perty ?

Thank you !Thank you !