The connexion of duality and causal properties for generalized free fields

It is shown that the time-slice axiom and the diamond property are equivalent for the generalized free field. If, in addition, there is a mass gap, duality is equivalent to either causality requirement. It is further shown that the local rings associated with certain space-time regions are factors in the case of causal generalized free fields with mass gap. Necessary and sufficient conditions for causality and duality and some examples for causal and acausal generalized free fields are also given.