Fuzzy Complex Projective Spaces and their Star-products

TL;DR

This paper constructs an explicit associative *-product for fuzzy complex projective spaces, generalizing previous fuzzy sphere results, and recovers classical algebra in the infinite limit.

Contribution

It introduces a new explicit *-product for fuzzy complex projective spaces, extending fuzzy sphere methods to higher-dimensional cases.

Findings

Derived an explicit *-product formula for fuzzy complex projective spaces

Showed the algebra reduces to classical functions in the infinite matrix limit

Expressed derivatives as matrix commutators on fuzzy spaces

Abstract

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