Tensor Products Of Non-unital Operator Systems

In this thesis, we study tensor products of non-unital operator systems, quotient objects and quotient maps in the category of non-unital operator systems as well as non-unital operator system structures on a non-unital function system.More precisely, we define the maximal tensor product Max and the minimal tensor product Min of non-unital operator systems. Through that, we study nuclearity of non-unital operator systems. We prove that the minimal tensor product is an injective tensor product and there are only very few (Min, Max)-nuclear non-unital operator systems. We then introduce the concept of reduced tensor products of non-unital operator system-s through unitalization and study the greatest reduced tensor product max0. We show that there is a strong connection between (Min,max0)-nuclearity and C*-nuclearity.On the other hand, we give the defintion of quotient object in the category of non-unital operator systems through regulization and prove that this approach is actually the same with unitalization. We then give the definition of complete NUOS-quotient map and prove that max0is a projective tensor product in some sense.Finally, we define the maximal and the minimal non-unital operator system struc-tures on a non-unital function system. These structures can also be obtained by using the maximal and minimal operator space structures endowed with certain matrix or-der structures. We find some relations between these structures and operator system structures on a function system.