If A is a commutative C*-subalgebra of linear bounded operator space B(H) on some Hilbert space H, so is the double commutant A. It follows from A is dense in A and the multiplication is continuous on each factor respectively, respect to the strong operator topology. 1) Can the assertion A is commutative be verified in an algebraic way? 2) Or, a more general question: If A is a commutative subalgebra of an algebra B, is the double commutant A of A also commutative?

Question

If A is a commutative C*-subalgebra of linear bounded operator space B(H) on some Hilbert space H, so is the double commutant A. It follows from A is dense in A and the multiplication is continuous on each factor respectively, respect to the strong operator topology.  1) Can the assertion A is commutative be verified in an algebraic way?  2) Or, a more general question: If A is a commutative subalgebra of an algebra B, is the double commutant A of A also commutative?

Elois Puryear · Accepted Answer

If A is commutative, you haveA′⊂A″ThenA′⊃A″And thenA″⊂A‴So A&#039;&#039; is contained in its commutant and is thus commutative.The above reasoning shows that all even higher commutants of A are commutative.