Mathematical Sciences: Structure of Vector-Valued Function Spaces and Non-Commutative Function Spaces

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9703789 Randrianantoanina The proposer intends to focus on two lines of research: (1) Banach space structures of C*-algebras and related spaces; (2) studies of vector-valued function spaces. In (1), he will study some connections between the two fields: Banach space theory and operator algebra theory. The primary goal of this direction of research is to show that some classical results from Banach space theory and harmonic analysis are still valid for the non-commutative setting. One important direction is the investigation of symmetric spaces of measurable operators associated with semi finite von Neumann algebras. The basic permanence question is whether or not a given property can be preserved from a given function space to its non commutative version. For instance, the classification of (non commutative) spaces of finite type (respectively of finite cotype) is not known; there is the question of which spaces have the Radon-Nikodym properties associated with Riesz-subsets of a given discrete group. Another direction is the study of non commutative Hardy spaces associated with maximal finite subdiagonal algebras of finite von Neumann algebras. For example, it is unknown if Szego's theorem from classical analysis is valid for the general non commutative setting. It is still open if the predual of such maximal subalgebras are of finite cotype. The same kind of investigation will also be considered for non commutative Hardy spaces and the non commutative disc algebras introduced by Popescu. Our knowledge of Banach space structures of non commutative C*-algebras (despite some recent intensive studies) is still less than satisfactory; the proposer also will tackle some of the questions that arise naturally from the classical case. For instance, the behavior of different classes of bounded operators on C*-algebras are far from being well understood. Several intermediate results, such as factorizations of absolutely summing operators and compactness of absolutely summing operators, were already obtained, hinting that in general non commutative C*-algebras should behave (in many ways) like their commutative counterparts. The main motivation for this part is that certain problems related to these algebras become more transparent when formulated in Banach space language. Part (2) can be viewed as close connections between measure theory and Banach space theory. Many questions remain unresolved on permanence properties preserved by Bochner spaces. The proposer will concentrate on properties of spaces containing classical function spaces, uniqueness of preduals, and strong regularity. The purpose of this project is to improve the understanding of the connections between different field of analysis: Banach space theory, harmonic analysis, and operator algebra theory. 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, mathematical physics, geometry, quantum mechanics), so it is important to view them from many different angles.