Some Problems On The Free Products Of Operator Algebras

The thesis consists of three chapters.Chapter 1 is the introduction. It mainly introduces the background of some problems involved in this thesis and lists some main results of this thesis.Chapter 2 gives a construction of the reduced free product of C~*-algebras and proves that it is equivalent to the construction given by D. Voiculeseu.Chapter 3 is devoted to the study of the free products of the operator spaces. We prove the university properties of the full free product of the operator spaces and the full free product of TROs. Then we define the full free product of completely contractive maps in operator spaces and the full free product of TRO homomorphisms, and show that these maps are also completely contractive or TRO homomorphism. Finally, we introduce the notions of the full amalgamated free product of the operator spaces and the full amalgamated free product of completely contractive maps, and give a concrete construction and some properties.