  We show that the one loop amplitudes of open and closed string theory in a
constant background two-form tensor field are characterized by an effective
string tension larger than the fundamental string tension, and by the
appearance of antisymmetric and symmetric noncommutativity parameters. We
derive the form of the phase functions normalizing planar and nonplanar tachyon
scattering amplitudes in this background, verifying the decoupling of the
closed string sector in the regime of infinite momentum transfer. We show that
the functional dependence of the phase functions on the antisymmetric star
product of external momenta permits interpretation as a finite wavefunction
renormalization of vertex operators in the open string sector. Using
world-sheet duality we clarify the regimes of finite and zero momentum transfer
between boundaries, demonstrating the existence of poles in the nonplanar
amplitude when the momentum transfer equals the mass of an on-shell closed
string state. Neither noncommutativity parameter has any impact on the
renormalizability of open and closed string theory in the Wilsonian sense. We
comment on the relationship to noncommutative scalar field theory and the UV-IR
correspondence.
