Literature status of bi-invariant mean restriction to closed subgroups

Determine whether the identity m_H(f|_H) = m_G(f) holds as a previously documented result in the literature, where G is a topological group, H ⊂ G is a closed subgroup, m_G and m_H denote the unique bi-invariant means on WAP(G) and WAP(H) respectively, and f ∈ WAP(G) is restricted to H. If it is known, identify authoritative references establishing this fact.

Background

The paper studies weakly almost periodic functions WAP(G) on a topological group G and their unique bi-invariant mean m, a key tool used throughout Section 2 and later in the development of the weakly mixing approximation property (WMAP) for tracial von Neumann algebras.

Immediately before Proposition \ref{sbgroup}, the authors note uncertainty about whether the specific restriction property of the bi-invariant mean to closed subgroups is already documented in the literature. They then provide a full proof of the proposition asserting m_H(f|_H) = m_G(f) for a closed subgroup H ⊂ G and f ∈ WAP(G).