Banach space structures of L^p-spaces and non-commutative Hardy spaces

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The proposed research will focus on problems concerning Banach space theory and its connections with operator algebras and operator theory. The main aim of this research is the investigation of structures of non-commutative L^p-spaces and non-commutative Hardy spaces. The basic permanence question in this direction of research is wether or a given property can be lifted from a given function space to its non-commutative version. One of the questions that will be considered is the classification of non-commutative spaces according to type and cotype. Another significant question is whether or not reflexive subspaces of preduals of von Neumann algebras have the fixed point property. Basic Banach space structure of the newly defined script-L^p-spaces and severalnon-commutative generalizations of Hardy spaces are at the center of this investigation. Another direction of research to be considered is the study of ideals of operators on C*-algebras.This proposal represents work of an interdisciplinary nature on mathematical analysis. Banach space theory, which is the main topic of this proposal, studies notions of distances on infinite dimensional vector spaces. It provides general framework for several fields of mathematics. The theory of function spaces played a crucial role in the development of Banach space theory for several decades. The current project studies a relatively new concepts of non-commutative analog of function spaces in which functions are replaced by operators. These spaces includes C*-algebras, preduals of von Neumann algebras among many others. C*-algebras turn out to be one of the mostimportant structures in mathematics. They have significant applications to other parts of sciences (for examples, geometry, mathematical physics and quantum mechanics), so it is important to consider them from many different point of view. In this case as Banach spaces.