Tensor Products Of Nest Subalgeba Of Nest Algebra

As an example of the type of translation which can be made,consider Tomita’s tensor product commutation formula of C*-algebras,Gilfeather,Hopenwasser and Larson[12]proposed the tensor product formula of nest algebras,and proved it.On the basis of the concept of nest subalgebras of von Neumann algebras,we raised the concept of nest subalgebras of nest algebras,and carry out the tensor product formula:AlgL1(?)AlgL2 = Alg(L1(?)L2).Where L1,and L2 is corresponding sublattice respectively.The formula is a promotion of Gilfeather,Hopenwasser and Larson’s result[12].On the basis of the definition of left slice map and right slice map,the Fubini product F(A,B)is important tool which has been given in this paper.Given anyσ-weakly closed subspaces B of B(K),if F(A,B)= A(?)B is satisfied,then we say A has property Sσ.Kraus proved that either lattice of L1 and L2 meet the condition which has property Sσ,the equation is identical.Zhang[13]carry out property πσ about von Neumann algebra which is weaker than property Sσ,and obtain the tensor products formula about nest subalgebras of von Neumann algebras.Learned from property Sσ and property πσ,then we give a new definition of property πN.Given any nest algebra B=AlgN of B(K),and A is a σ-weakly closed subspace of B(H),if F(A,B)=A(?)B is satisfied,then we say A has property πN.Then we find nest subalgebra A = AlgA0 L of nest algebra.A0 = AlgN has property πN on condition L(?)N.Further,for any nest subalgebra A = Alg A0 L of nest algebra A0=AlgN,A as property πN,B also has propertyπN,then we have F(A,B)= A(?)B.So far,we can show that the tensor products formula of nest subalgebra of nest algebra:Alg A1 L1(?)AlgA2L2=AlgA1(?)A2(L1(?)L2)is valid,where the lattices L1 and L2 is nest of nest algebras A1=AlgN1 and A2 = AlgN2 separately,moreover Li(?)Ni,i= 1,2.With the introduction of new concepts and new methods in this paper,we get a better result than Gilfeather,Hopenwasser and Larson[12].