A very strong difference property for semisimple compact connected lie groups

Let G be a topological group. For a function f: G → ℝ and h ∈ G, the difference function Δ _h f is defined by the rule Δ _h f(x) = f(xh) − f(x) (x ∈ G). A function H: G → ℝ is said to be additive if it satisfies the Cauchy functional equation H(x + y) = H(x) + H(y) for every x, y ∈ G. A class F of real-valued functions defined on G is said to have the difference property if, for every function f: G → ℝ satisfying Δ _h f ∈ F for each h ∈ G, there is an additive function H such that f − H ∈ F. Erdős’ conjecture claiming that the class of continuous functions on ℝ has the difference property was proved by N. G. de Bruijn; later on, F. W. Carroll and F. S. Koehl obtained a similar result for compact Abelian groups and, under the additional assumption that the other one-sided difference function ∇ _h f defined by ∇ _h f(x) = f(xh) − f(x) (x ∈ G, h ∈ G) is measurable for any h ∈ G, also for noncommutative compact metric groups. In the present paper, we consider a narrower class of groups, namely, the family of semisimple compact connected Lie groups. It turns out that these groups admit a significantly stronger difference property. Namely, if a function f: G → ℝ on a semisimple compact connected Lie group has continuous difference functions Δ _h f for any h ∈ G (without the additional assumption concerning the measurability of the functions of the form ∇ _h f), then f is automatically continuous, and no nontrivial additive function of the form H is needed. Some applications are indicated, including difference theorems for homogeneous spaces of compact connected Lie groups.