  We propose an abstract framework describing energy-renormalized Hamiltonians
in terms of local algebras. Within the framework, we examine the positivity
improvingness of the semigroup generated by the renormalized Hamiltonian. As
examples, we discuss the renormalized Nelson Hamiltonian and the renormalized
Nelson Hamiltonian at fixed total momentum. The characteristic features of our
approach are as follows:(i) in contrast to the probabilistic approach in the
Schr\"odinger representation, our method works well in the Fock representation;
(ii) the method covers the massless case.
