Trace inequalities for functions of several variables
by
Gert K. Pedersen
University of Copenhagen
Coauthors: Frank Hansen (University of Copenhagen)

Let f be a function of n real variables defined on a cube I=I₁× ... ×I_n in Rⁿ and let \tau be a trace on a C^*-algebra A. If x_k = (x_1k, ..., x_nk) for 1 k m is a family of commuting self-adjoint elements in A, i\.e\. [x_ik, x_jk]=0, with spectra in I (so sp(x_ik) I_k) and if (a₁, ..., a_m) is a left unital m-tuple in M(A), i\.e\. \sum_k=1^m a^*_ka_k=1, we show that

\tau æ è f æ è m å k=1  a*kxk ak ö ø ö ø <= \tau æ è m å k=1  a*kf(xk)ak ö ø ,

provided that the elements y_i = \sum_k=1^m a^*_kx_ika_k for 1 <= i <= n form a commuting n-tuple in A.

This represents the multi-variable version of jensen's trace inequality. In the special case where the a_k's are scalars we let \lambda_k = a^*_ka_k to obtain a (commutative) convex combination of the tuples (x_k). Now the inequality reads:

\tau æ è f( m å k=1  \lambdakxk) ö ø <= \tau(\lambdakf(xk)),

provided that the convex combination of the tuples again form a commutative tuple.

The surprising fact is that we can obtain convexity estimates on a set (of commuting n-tuples of self-adjoint operators) that is only a partially convex set. For n=2 this means that we have convexity estimates on the set of normal elements in A.

Date received: June 23, 2002