  The wedge states form an important subalgebra in the string field theory. We
review and further investigate their various properties. We find in particular
a novel expression for the wedge states, which allows to understand their star
products purely algebraically. The method allows also for treating the matter
and ghost sectors separately. It turns out, that wedge states with different
matter and ghost parts violate the associativity of the algebra. We introduce
and study also wedge states with insertions of local operators and show how
they are useful for obtaining exact results about convergence of level
truncation calculations. These results help to clarify the issue of anomalies
related to the identity and some exterior derivations in the string field
algebra.
