Some Problems Of The Free Product Of C～*-Algebras

The free product of C*-algebras, defined by D.Avitzour and D.V.Voiculescu independently in 1980s, is one of the important research subjects in operator algebra theory. This thesis concerns some problems of the free products of C*-algebras, and is composed of four chapters.In Chapter 1 we introduce the notions and definitions of the C*-algebras and the free product of C*-algebras, and list the main results in this thesis.In Chapter 2 we show that the semi-norm || · ||_o defined by ||a||_o = sup{||Ï€(a)|| | Ï€ is the *-representation of A₁ *_a A₂ *a ··· *_a A_n }((?)a∈ G A₁ *_a A₂ *_a ···*_a A_n) is indeed a C*-norm. And also give a counterexample for establishing that the free products of the faithful *-representations need not be faithful.In Chapter 3 we give some sufficient and necessary conditions on which the linear functional on the *-algebraic free products of C*-algebras, and respectively, the dense *- subalgebras, extend to be the states of the universal C*-free product.In Chapter 4 we define the full amalgamated free products of the completely positive maps, and give some reduced properties of the full amalgamated free products of the completely positive maps.