Quantum Probabilistic Methods in Operator Spaces and Applications

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The starting point of this project on 'Lp estimates in noncommutative probability and analysis' are recent results and techniques from noncommutative martingales inequalities obtained by Pisier/Xu, Randrianantoanina, Junge and Junge/Xu. This new insight enables us to show a noncommutative analog the maximal ergodic theorem for completely positive maps and study square function inequalities in this setting (joint work with LeMerdy/Xu). The formulation and properties are motivated by the theory of Operator Spaces. On the other hand martingale inequalities are crucial in understanding independent, indiscernable and exchangeable sequences in noncommutative Lp spaces and properties of almost uniform convergence. Martingale inequalities and noncommutative probability are also fundamental tools in analyzing the operator space OH and its realization in the predual of type III von Neumann algebras. These techniques are similar to those used by Pisier/Shlyaktenko in proving the noncommutative version of Grothendieck's inequality. Surprising the analog of the 'little Grothendieck inequality' in the context of operator spaces only holds up to a logarithmic factor.Quantum mechanics and Heisenberg's uncertainty principle and mathematical models realizing these phenomena changed not only our perception of the world but also the mathematical discipline. Many noncommutative (=quantum) generalizations of classical mathematical theories for example the theory of quantum groups and noncommutative (=quantum) probability theory. Very interesting new phenomena and difficulties arise when adopting classical concepts to this noncommutative framework. Noncommutative measure theory and the theory of von Neumann algebras provide plenty of examples of genuinely new phenoma. Indeed, von Neumann's motivation for his work on operator algebras (now called von Neumann algebras) was to provide a good mathematical foundation for quantum mechanics. In this tradition the theory of Operator Spaces provides the right language for quantizing Banach spaces, a notion developed to describe the spaces of solutions of differential equations. For example, using this language it is now possible to talk about the expected exit time for a noncommutative domain although we can never see the "points" of this domain. As a long term perspective these mathematical theories provide new features which may be used to understand phenomena in physics and other natural sciences.