Non-commutative Lp-spaces and their Connection to Probability and Operator Spaces

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AbstractJungeThe aim of this research is the investigation of the followingdifferent aspects of non-commutative spaces of p-integrable functions. If p is 1 such a space is the predual of von Neumann algebra and reflects important properties of the underlying operator algebra. We recall, that it is still open whether preduals of von Neumann are finitely represented in the space of trace class operators. Here, we focus on isometric characterization of finite dimensional spaces embedding into the predual of a von Neumann algebra and its connection to the theory of Lie-algebras and (non-commutative) stochastical processes. The investigation of the latter uses martingale inequalities based on recent progress by Pisier and Xu. We are interested in the non-commutative version of the Rosenthal/Burkholder inequality and Doob's maximal inequality. Maximal inequalities are also known as a useful tool in (stochastical) analysis. The more recent theory of operator spaces delivers the right framework for these investigations and reveals surprising properties of the non-commutative space of p-integrable functions associated to free groups. Non-commutative probability provides one possible framework for the probabilistic viewpoint in quantum mechanics. This theorycombines fundamental concepts of algebraic nature with analyticinsight and methods with roots in calculus. The non-commutative analogue for the spaces of p-integrable functions has a long tradition in the theory of operator algebras and provides a fruitful framework for understanding classical tools in probability. It is most challenging to reveal or overcome substantial differences between the commutative and non-commutative theory. This area enables the interaction between different streams inside the mathematical community and mathematical physics. This kind of interaction is one of the most important resources for new development in mathematics