Diffeomorphisms and approximate invariants on fuzzy sphere

We consider how diffeomorphisms act on the configuration space of matrices in the matrix regularization. We construct the matrix regularization in terms of the Berezin-Toeplitz quantization and define diffeomorphisms on the space of matrices by using this quantization map. For the case of the fuzzy $S^2$, we explicitly construct the matrix version of holomorphic diffeomorphisms on $S^2$. We also propose three methods of constructing approximate invariants on the fuzzy $S^2$. These are exactly invariant under unitary similarity transformations and only approximately invariant under the general diffeomorphisms.