Thomas Unger: Extending orderings and valuations to algebras with involution

Central simple algebras with involution are a very active area of current research in algebra with many connections to other parts of mathematics. In the last number of years, Vincent Astier and myself have been developing the “real theory” of algebras with involution, which means investigating these objects in the presence of orderings on the base field. A natural question that arises (and that is motivated by the theory of positive semidefinite matrices and results on signatures of hermitian forms), is wether orderings can be extended from the base field to the algebra, and how much of the Artin-Schreier theory of ordered fields carries over. Another question is to see how much of the Baer-Krull theory (the behaviour of orderings under real places) carries over. Pretty soon after these results were established for fields, extensions in the noncommutative direction were attempted, some of which more successful than others, but none really ticking all the boxes. In my talk I will discuss our work on “positive cones” and their canonically associated Tignol-Wadsworth gauges in this context, which so far already has produced several promising results.