  Recently Lewis Bowen introduced a notion of entropy for measure-preserving
actions of countable sofic groups admitting a generating measurable partition
with finite entropy; and then David Kerr and Hanfeng Li developed an
operator-algebraic approach to actions of countable sofic groups not only on a
standard probability space but also on a compact metric space, and established
the global variational principle concerning measure-theoretic and topological
entropy in this sofic context. By localizing these two kinds of entropy, in
this paper we prove a local version of the global variational principle for any
finite open cover of the space, and show that these local measure-theoretic and
topological entropy coincide with their classical counterparts when the acting
group is an infinite amenable group.
