  The open string on the plane-wave limit of $dS_n\times S^n $ with constant
  $B_2$ and dilaton background fields is canonically quantized. This entails
solving the classical equations of motion for the string, computing the
symplectic form, and defining from its inverse the canonical commutation
relations. Canonical quantization is proved to be perfectly suited for this
task, since the symplectic form is unambiguously defined and non-singular. The
string position and the string momentum operators are shown to satisfy
equal-time canonical commutation relations. Noticeably the string position
operators define non-commutative spaces for all values of the string
world-sheet parameter $\sig$, thus extending non-commutativity outside the
branes on which the string endpoints may be assumed to move. The Minkowski
spacetime limit is smooth and reproduces the results in the literature, in
particular non-commutativity gets confined to the endpoints.
