A note on blocks with abelian defect groups

A recent result by H. Meyer shows that, for a field F of characteristic p>0 and a finite group G with an abelian Sylow p-subgroup, the F-subspace Zp′FG of the group algebra FG spanned by all p-regular class sums in G is multiplicatively closed, i.e. a subalgebra of the center ZFG of FG. Here we generalize this result to blocks. More precisely, we show that, for a block A of a group algebra FG with an abelian defect group, the F-subspace Zp′A:=A∩Zp′FG is multiplicatively closed, i.e. a subalgebra of the center ZA of A. We also show that this subalgebra is invariant under perfect isometries and hence under derived equivalences.