  A criterion for M\"uger centralizer of a fusion subcategory of a braided
non-degenerate fusion category is given. Along the way we extend some
identities on the space of class functions of a fusion category introduced by
Shimizu in \cite{scalg}. We also show that in a modular tensor category the
product of two conjugacy class sums is a linear combination of conjugacy class
sums with rational coefficients.
