Jensen's inequality for spectral order and submajorization

Abstract

Let A be a C ∗ -algebra and ϕ : 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 ( a ) ) ⩽ ϕ ( 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.