  We establish extremality of Riemannian metrics g with non-negative curvature
operator on symmetric spaces M=G/K of compact type with rk(G)-rk(K)\le 1. Let
g' be another metric with scalar curvature k', such that g'\ge g on 2-vectors.
We show that k'\ge k everywhere on M implies k'=k. Under an additional
condition on the Ricci curvature of g, k'\ge k even implies g'=g. We also study
area-non-increasing spin maps onto such Riemannian manifolds.
