Matthias Schötz (University of Würzburg) will speak in the geometry seminar on the 13th of February. The talk will take place in the Salle de Profs (9th floor, NO) at 13.30pm. Matthias’ title is “From non-formal, non-C* deformation quantization to abstract O*-algebras” and the abstract is below.
Starting with any hilbertisable locally convex space V (i.e. locally convex space whose topology can be described by inner products), one can construct its usual deformations by means of exponential star products (like Moyal and Wick star product) on the commutative *algebra of polynomial functions over V, and finds that there is a unique coarsest topology on the deformed *algebras making all deformed products, all evaluation functionals and the *involution continuous. If V has finite dimension, then from this one can also derive a convergent star product on the hyperbolic disc by symmetry reduction.
The resulting deformed *algebras have some more nice properties, e.g. in the flat case it is possible to incorporate elements Q,P having canonical commutation relations [Q,P] = i and to exponentiate these elements in the completion of the algebra, and on the hyperbolic plane the usual su(1,n)-action is an inner action on the deformed *-algebra. Nevertheless, their topologies are far from being C*, yet not even submultiplicative. So the question arises, which of the properties that make C*-algebras attractive as candidates for observable algebras in physics carry over to these constructions or similar ones. The notion of an abstract O*-algebra might provide a suitable framework to examine these problems: The idea is to focus more on the properties of the ordering on a *algebra coming from a suitable set of positive linear functionals, which e.g. allows to study properties of pure states in detail, and could eventually lead to a spectral theorem for *algebras of unbounded operators by applying the Freudenthal spectral theorem for lattice ordered vector spaces.