Trace Inequalities in Several Variables

Abstract: Let f be a convex function of n variables, and use it on n-tuples of commuting self-adjoint operators (inside a given C*-algebra A) to create an operator function . The trace of this function is then convex - insofar as it is defined. (Here trace refers to any trace on A.) We even have that the operator function respects noncommutative convex combinations, where scalar multiples are replaced by products a*xa. The main problem is to understand the domain of definition, i.e. sets of commuting n-tuples of self-adjoint elements which are "compatible" with each other in the sense that the line between any two tuples also consists of pairwise commuting elements.