  We derive an explicit expression for an associative star product on
non-commutative versions of complex Grassmannian spaces, in particular for the
case of complex 2-planes. Our expression is in terms of a finite sum of
derivatives. This generalises previous results for complex projective spaces
and gives a discrete approximation for the Grassmannians in terms of a
non-commutative algebra, represented by matrix multiplication in a
finite-dimensional matrix algebra. The matrices are restricted to have a
dimension which is precisely determined by the harmonic expansion of functions
on the commutative Grassmannian, truncated at a finite level. In the limit of
infinite-dimensional matrices we recover the commutative algebra of functions
on the complex Grassmannians.
