Research On Partial Order Structures On Operator Algebras

Partial order structure is an important area in functional analysis and has wide applications in set theory,discrete mathematics,graph theory,lattice theory,statistics,information and computing science.The member relationship of a set can be reflected by partial order,so it is of great significance when we investigate some special sets systematically.The research on partial order structures on operator algebras has always been hot questions explored by scholars.The characterization of partial order structures may reveal the inherent properties of operator algebra and present new tools and viewpoint for the research of the essential invariant of operator algebras.The important features of partial order structures mainly content order automorphism,the boundedness and hereditary properties of partial order.The order automorphism of operator algebra reflect that partial order can be a stiffness features of operator algebras.The characterization of bounds may reflect the lattice properties of operator algebra more specifically.Moreover,the hereditary subspace may reflect the structure and properties of operator algebra in the viewpoint of the hereditary properties of partial order.In this paper,we consider two partial order on operator algebras,diamond order and star partial order.We study the diamond order on the von Neumann algebra B(H)of all bounded linear operators on a Hilbert space H and consider the problem of order automorphism on the unit interval under the diamond order.We give a type decomposition of operators with respect to star partial order and determine all automorphisms on the poset of type 1 operators.Moreover,we extend these two partial orders to the non-commutative Lp space associated with semi-finite von Neumann algebras and consider the problem of bounds and hereditary subspaces.First,we reveal the relationship between the the unit interval of diamond order and the poset of products of two projections.Using the canonical factorization of the operator of products of two projections,we give some fundamental properties of this poset.We define a special class and equivalence relation and establish a corresponding relation between them and the rank one operators in the poset of products of two projections.We characterize the order automorphisms on the unit interval under the diamond order.Second,we define the type 1 and type 2 operators in B(H)with respect to star partial order and prove that every operator can be represented as a sum of these two type operators uniquely.By using this type decomposition in the study of star order automorphism,we determine all automorphisms on the poset of type 1 operators.As a consequence,we characterize continuous automorphisms on B(H).Last,we consider these two partial orders on the non-commutative Lp space associated with semi-finite von Neumann algebras and study the problem of star infimum and supremum and the diamond minimal upper bound.We prove that under the star partial order,if a subset has an upper(res.lower)bound,then it must has supremum(res.infimum).We also prove that a subset with an upper bound must have a minimal upper bound in Lp(M)under the diamond order in the case of finite von Neumann algebra M.However,we give an example and show that this result may fail if M is not finite.Moreover,we characterize the forms of all norm closed hereditary subspaces in Lp(M)under these two partial orders.