Banach-Kantorovich C*-algebras and zero-two laws for positive contractions

Abstract

In this thesis, we study C^*-algebras over Arens algebras. Moreover, we consider C^*-algebra of sections and will prove that C^*-algebra over L^? is isometrically *-isomorph to C^*-algebra L^? (?,X). Furthermore, we investigate the state space of C^*-algebras over L^?. We also study dominated operators acting on Banach-Kantorovich L_p-lattices. Further, using the methods of measurable bundles of Banach-Kantorovich lattices, we prove the strong zero-two law for the positive contractions of the Banach-Kantorovich lattices L_p (?,m). After that, we illustrate an application of the methods used in previous study to prove a result related to dominated operators. Thereafter, we collect some necessary well-known facts about non-commutative L_1-spaces. Then we prove an auxiliary result about dominant operators. Next, we prove a generalized uniform "zero-two" law for multi-parametric family of positive contractions of the non-commutative L_1-spaces. Furthermore, we recall necessary definitions about L_1 (M,?) – the non-commutative L_1-spaces associated with center valued traces and we show auxiliary result about the existence of the non-commutative vector-valued lifting. Finally, we prove that every positive contraction of L_1 (M,?) can be represented as a measurable bundle of positive contractions of non-commutative L_1-spaces, and this allows us to establish a vector- valued analogue of the uniform "zero-two" law for positive contractions of L_1 (M,?).