  In this paper we study the midpoint structure of the algebra of open strings
from the standpoint of the operator/Moyal formalism. We construct a split
string description for the continuous Moyal product of hep-th/0202087, study
the breakdown of associativity in the star algebra, and identify in infinite
sequence of new (anti)commutative coordinates for the star product in in the
complex plane. We also explain how poles in the open string
non(anti)commutativity parameter correspond to certain ``null'' operators which
annihilate the vertex, implying that states proportional to such operators tend
to have vanishing star product with other string fields. The existence of such
poles, we argue, presents an obstruction to realizing a well-defined
formulation of the theory in terms of a Moyal product. We also comment on the
interesting, but singular, representation $L_0$ which has appeared prominently
in the recent studies of Bars {\it et al}.
