Banach Space Structures of Non-commutative L^p-spaces and Non-commutative Martingale Inequalities

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This research project will focus on problems dealing with Banach space theory and its connections with operator algebras and operator theory. The main directions to be considered are (1) aspects of non-commutative spaces of p-integrable functions/operators, (2) non-commutative Hardy spaces, and (3) non-commutative martingale inequalities. If p is equal to 1, the non-commutative space of integrable operators is the predual of von Neumann algebra and reflects many important properties of the underlying algebras. The basic permanence question in this direction of research is whether or not a given property can be lifted from a given function space to its non-commutative counterpart. An important direction to be considered is the study of different structural properties of reflexive subspaces of preduals of von Neumann algebras. For instance, it is still unknown if such spaces have the fixed point property for non expansive maps. For the case of Hardy spaces, the primary goal is to develop and extend tools from harmonic analysis to the non-commutative settings. In a somewhat different direction the ongoing research on non-commutative martingales will be considered. One of the main goals is to find the appropriate quantization of different classical inequalities, and apply these results to C*-algebras and von Neumann algebras, as well as to some other related areas.This research 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 concept of non-commutative analogue 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 most important 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. Non-commutative probability provides one possible framework for a probabilistic viewpoint in quantum mechanics.