An invariant state satisfying the Kubo-Martin-Schwinger condition is studied. It is shown that the decomposition of a given state into extremal invariant states yields states satisfying the KMS boundary condition if and only if the cyclic representation associated with the given state is ??-abelian, and that, if this is the case, the decomposition coincides with the standard central decomposition. The structure of the cyclic representation when it is non ?7-abelian is analyzed and typical examples are given. One of the examples gives a case where the cyclic representation is G-abelian but not Ty-abelian.