  Perturbative Algebraic Quantum Field Theory (pAQFT) is based upon formal
power series valued in spaces of functionals. This is usually done with
microcausal functionals, which are defined using microlocal analysis and
motivated by propagation of singularities. In this paper, we prove that the
class of microcausal functionals is not closed under the Peierls (Poisson)
bracket by showing that a Peierls bracket of regular functionals can fail to be
smooth. Consequently, microcausal functionals are not a suitable basis for
pAQFT. To remedy these issues, we introduce the class of equicausal
functionals. We show that this class contains the local functionals and that it
closes under the star-product and Peierls bracket. Furthermore, we prove the
time-slice axiom for equicausal functionals, using a chain homotopy. The class
of microcausal functionals is not closed under this chain homotopy, which
strongly suggests that the class of microcausal functionals does not fulfill
the time slice axiom.
