here is a vague question:
I recognized that adjoints of operators and commutants of non-degenerate self-adjoint algebras share similar features, e.g. the double adjoint of a closable, densely defined operator is the closure of this operator. The double commutant of an non-degenerate self-adjoint algebra is the strong/weak closure of it (double commutant theorem). It seems to me, that this property of double(something) implies closure is a recurring feature and I was wondering whether they are somehow related in a maybe category-theory kind of sense?
Thanks