Auburn Algebra Seminar

Let \(f : (0,\infty) \to (0,\infty)\) be a nontrivial normalized operator monotone function and set \(s = f'(1)\). We introduce the alternative-mean trace functional

on the positive definite cone. We prove that \(\widehat{\Phi}_f\) is a quantum divergence in the sense of Bhatia–Gaubert–Jain whose diagonal Hessian induces a positive multiple of the Bures–Wasserstein Riemannian metric. We also establish the sharp comparison

We further establish several equivalent conditions characterizing those \(f\) for which \(\widehat{\Phi}_f\) satisfies data processing under all positive trace-preserving maps; in particular, diagonal pinching on faithful qubit states is a complete test. For the associated barycenter problem, we prove existence of closed-cone minimizers, derive a first-order equation for interior critical points, and give criteria ensuring uniqueness and positive definiteness. For the power family \(f(x) = x^t\), we determine exactly when every finite data set in the positive definite cone admits a unique closed-cone minimizer, which is then necessarily positive definite.