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In this academic year the researcher continued the work on extending Loewner’s theorem to several variables. The first result is an new LMI characterization of operator concave and monotone functions. This characterizes the functions as unique extremal solutions of linear matrix inequalities over some auxiliary Hilbert space. This result was presented at the ’Recent developments in operator algebras’ workshop in RIMS.
Later using this LMI representation an exact solution formula was established which seems to be the key to obtain the analytic continuation part of Loewner’s theorem. This part is still being carried out at the moment. As a by-product an infinite dimensional version of the theory of matrix convex sets and non-commutative Hahn-Banach theorems were established as key tools.
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