  A notion of local algebras is introduced in the theory of causal fermion
systems. Their properties are studied in the example of the regularized Dirac
sea vacuum in Minkowski space. The commutation relations are worked out, and
the differences to the canonical commutation relations are discussed. It is
shown that the spacetime point operators associated to a Cauchy surface satisfy
a time slice axiom. It is proven that the algebra generated by operators in an
open set is irreducible as a consequence of Hegerfeldt's theorem. The light
cone structure is recovered by analyzing expectation values of the operators in
the algebra in the limit when the regularization is removed. It is shown that
every spacetime point operator commutes with the algebras localized away from
its null cone, up to small corrections involving the regularization length.
