The Lattice Properties And Hereditary Subspaces Of The *-partial Order In A Von Neumann Algebra

The order relation holds a certain position on semigroups.The lattice properties and hereditary subspaces of partial order are two important research direction of the operator algebras.In this thesis,we make further research into star partial order on a von Neumann algebra,and mainly considered the lattice properties and star partial order-hereditary subspaces.Let H be a complex Hilbert space and B(H)the algebra of all bounded linear operators on H Let M(?)B(H)be a von Neumann algebra and "≤" the star partial order in M,that is,for A,B ∈ M,then we say that A<B if A*A = A*B and A A*= BA*.We have the following content:In the first part,we study the lattice properties with respect to the star partial order on a von Neumann algebra.First of all,the paper gives the representation of the supremum and infimum of two operators A and B on B(H)by matrix decom-position.Then,on the basis of this,it is proved that the supremum and infimum of a subset in M with respect to the star partial order are the same as in B(H)by these conditions and conclusions.In the second part,the star partial order-hereditary subspaces on a von Neu-mann algebra M is taken into consideration.By hereditary subspace,that is,let"≤" be a partial order on M and A a subspace of M,if A ∈M,B∈ A,and A ≤ B imply A ∈M,then we say that M is a hereditary subspace with respect to the partial order "≤".The paper gives the definition of a star partial order-hereditary subspace in M,and describe the characterizations of star partial order-hereditary subspaces in M.It is proved that a weak*closed subspace A in M is hereditary subspace with respect to the star partial order,if and only if there is a unique pair of non-zero projections E and F which have the same central carrier in M such that A=EMF.