Fuzzy complex projective spaces and their star-products

We derive an explicit expression for an associative ∗-product on the fuzzy complex projective space, CPN−1F. This generalises previous results for the fuzzy 2-sphere and gives a discrete non-commutative algebra of functions on CPN−1F, represented by matrix multiplication. The matrices are restricted to ones whose dimension is that of the totally symmetric representations of SU(N). In the limit of infinite-dimensional matrices we recover the commutative algebra of functions on CPN−1. Derivatives on CPN−1F are also expressed as matrix commutators.