  The standard reduced bar complex B(A) of a differential graded algebra A
inherits a natural commutative algebra structure if A is a commutative algebra.
We address an extension of this construction in the context of E-infinity
algebras. We prove that the bar complex of any E-infinity algebra can be
equipped with the structure of an E-infinity algebra so that the bar
construction defines a functor from E-infinity algebras to E-infinity algebras.
We prove the homotopy uniqueness of such natural E-infinity structures on the
bar construction.
  We apply our construction to cochain complexes of topological spaces, which
are instances of E-infinity algebras. We prove that the n-th iterated bar
complexes of the cochain algebra of a space X is equivalent to the cochain
complex of the n-fold iterated loop space of X, under reasonable connectedness,
completeness and finiteness assumptions on X.
