Jensen's inequality for spectral order and submajorization

Abstract

Let A be a C*-algebra and φ{symbol} : A → L (H) be a positive unital map. Then, for a convex function f : I → R defined on some open interval and a self-adjoint element a ∈ A whose spectrum lies in I, we obtain a Jensen's-type inequality f (φ{symbol} (a)) ≤ φ{symbol} (f (a)) where ≤ denotes an operator preorder (usual order, spectral preorder, majorization) and depends on the class of convex functions considered, i.e., monotone convex or arbitrary convex functions. Some extensions of Jensen's-type inequalities to the multi-variable case are considered.