On Some Free Products Of Operator Spaces And Ternary Rings Of Operators

The free product theory of the C~*-algebras and von Ncumann algebras has become one of the important research subjects in the theory of operator algebras. The free products of von Ncumann algebras was introduced by W.M.Ching in 1973.Later,it was generalized independently to C~*-algcbras by Voiculcscu and Avitzour.Since then,plenty of work had been done.Recently,the concept of free products was generalized to operator spaces and ternary rings of operators(or simply TRO)for the first time by MingChu Gao,who gave the full free produet construction and the reduced free product construction for operator spaces,and gave the TRO-free product construction for ternary rings of operators.This thesis is devoted to study the free products of operator spaces and ternary rings of operators.It consists of two chapters.In chapter 1,we mainly study the free products of operator spaces.Using the universal free product of the free C~*-algebras of operator spaces,we introduce the definition of universal free produet of operator spaces and give its construction, and then show that,like the universal free products of C~*-algcbras,the free product of operator spaces also satisfies the universal property and is independent of the representations of the original operator spaces.We also show that the free C~*-algcbra of the universal free product of two operator spaces is *-isomorphic to the universal free product of the free C~*-algcbras of the two operator spaces.In chapter 2,we mainly study the free products of ternary rings of operators. Using the universal free product of the linking C~*-algcbras of ternary rings of operators,we introduce the definition of TRO-univcrsal free product and give its construction,which is provcd to satisfy thc univcrsal property and bc indcpcndent of the represcntations of the original TRO's.We also show that thc linking C~*-algcbra of the TRO-univcrsal free product of two TRO's is~*-isomorphic to thc universal free product of the linking C~*-algcbras of thc two TRO's.In addition, inspircd by thc concept of full amalgamated frcc product of C~*-algebras, by using thc full amalgamated free product of thc linking C~*-algcbras of ternary rings of operators,we introduce the definition of TRO-full amalgamatcd free product,and give its construction,which is provcd to satisfy the univcrsal propcrty. We also show that thc linking C~*-algcbra of the TRO-full amalgamated free product of two TRO's is *-isomorphic to thc full amalgamated free product of thc linking C~*-algcbras of thc two TRO's.