Do all positive elements in a C* algebra commute?

Question

Let there be A, B positive elements in some C* algebra.
Now there holds: $$ AB = (AB)^* = B^*A^* = BA $$ since the product of A and B again gives a positive element, correct ?

Answer 1

No, it is not true that all positive elements of a $C^*$ algebra commute. The problem with your argument is that it is not true that the product of two positive elements in a $C^*$ algebra must be positive.