Fuzzy Complex Projective Spaces and their Star-products

We derive an explicit expression for an associative ∗-product on the fuzzy complex projective space, CPN−1 F . This generalises previous results for the fuzzy 2-sphere and gives a discrete non-commutative algebra of functions on CPN−1 F , 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−1 F are also expressed as matrix commutators