Criterion for the commutant of a quantized field to be algebraically closed

A necessary and sufficient condition is formulated for the commutant [4–5] of a given quantized field to be algebraically closed (i.e., closed with respect to algebraic operations). If the field satisfies the usual Wightman axions, the assumption that its commutant is a $^*$ algebra implies that this $^*$ algebra is abelian.