Jensen's Trace Inequality in Several Variables

Abstract

For a convex, real function f, we present a simple proof of the formula [Formula: see text] valid for each tuple (x1,…, xm) of symmetric matrices in [Formula: see text] and every unital column (a1,…, am) of matrices, i.e. [Formula: see text]. This is the standard Jensen trace ine-quality. If f ≥ 0 it holds also for the unbounded trace on [Formula: see text], where [Formula: see text] is an infinite-dimensional Hilbert space. We then investigate the more general case where τ is a densely defined, lower semi-continuous trace on a C*-algebra [Formula: see text] and f is a convex, continuous function of n variables, and show that we have the inequality [Formula: see text] for every family of abeliann-tuples [Formula: see text], i.e. tuples of self-adjoint elements in [Formula: see text] such that [xik, xjk] = 0 for all i, j and k, where 1 ≤ k ≤ m, and every unital m-column (a1,…, am) in [Formula: see text], provided that the elements [Formula: see text] also form an abelian n-tuple. We even establish this result for weak* measurable, self-adjoint, abelian fields (xit)t∈T, 1 ≤ i ≤ n, i.e. [xit, xjt] = 0 for all i, j and t, and weak* measurable, unital column field (at)t∈T in [Formula: see text] paired with any trace or trace-like functional φ, i.e. one that contains the n-tuple (presumed abelian) with elements [Formula: see text] in its centralizer. This takes the form of the inequality [Formula: see text] We also study functions of n variables that are monotone increasing in each variable, and show in two important cases that [Formula: see text] whenever [Formula: see text] and [Formula: see text] are abelian n-tuples with xi ≤ yi for each i and φ is a trace or a trace-like functional.