A note on factoring of positive definite functions on semigroups

The problem of characterizing those semigroups of class M such that every positive definite function factors via the greatest C-separative *-homomorphic image is shown to be equivalent to several other problems, including that of characterizing flat semigroups. If the latter could be solved then the problem of characterizing perfect semigroups would be reduced to characterizing quasi-perfect semigroups.