Relative Amenability Of Finite Von Neumann Algebras

In this paper,we study some properties of finite von Neumann algebras under the amenable inclusions.We prove that the inclusion N(?)M is amenable implies the identity map on M has an approximate factorization through N(?)Mm(C)via trace preserving normal unital completely positive maps,which is a generalization of a result of Haagerup.We also prove two permanence properties for amenable inclusions.One is weak Haagerup property,the other is weak exactness.In Chapter 3,we study the amenable inclusion of a finite von Neumann algebra M and its von Neumann subalgebra N.In the first part of this section,we obtain a characterization of amenable inclusions.We first give a description of this amenable inclusion an approximately factorization using normal bounded completely positive maps.Then,we make a rotation of cer-tain element in Mm(C)(?)N to prove that the amenable inclusion of finite von Neumann algebras N c M implies that the identity map on M can be approximated by normal unital completely positive maps.As the main result of this part,we prove a result which generalizes a result of Haagerup[1].Let M be a finite von Neumann algebra(resp.a type ?1 factor),N(?)M be a?1 factor(resp.N(?)M be a von Neumann subalgebra having atomic parts)and the inclusion N(?)M be amenable.Then the identity map on M can be approximated by normal unital trace preserving completely positive maps.Since unital completely positive maps are clearly to be completely bounded maps,our main results may serve us as a tool to obtain some results of completely bounded maps too.In Chapter 4,we show some applications of our main results.In[2],Popa asked if M is a finite von Neumann algebra,N(?)M is a von Neumann subalgebra,the inclusion N(?)M is amenable,and N has Haagerup property,does M also have Haagerup property?Bannon and Fang in[3]answered that question affirmatively from the point of view of correspondence.As the first application of our main result,we give that question a new proof for the case N(?)M a?1 factor in the frame of completely positive maps.We also prove that under our assumptions,M can inherit some properties of its subalgebra N.The properties are weak Haagerup property and weak exactness.