# 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.