Spectral representation of local symmetric semigroups of operators over topological groups

We consider local symmetric semigroups of Hilbert space operators. For an open semigroup G in some topological group and a dense subsemigroup G' of G, these are semigroups of unbounded selfadjoint operators (H(t)) _t∈G' that admit local continuous extensions to open subsets of G. We study the possibility to continuously extend H(·) to a semigroup of selfadjoint operators defined for all t ∈ G in several settings. Integral representation formulae for the extended semigroups (H(t)) _t∈G by means of real characters of G are established. Our proofs rely on graph limits of selfadjoint operators, commutativity of unbounded operators and semigroup techniques, among others.