Commutativity of Operators

For two bounded positive linear operators a, b on a Hilbert space, we give conditions which imply the commutativity of a, b. Some of them are related to well-known formulas for indefinite elements, e.g., (a+b) = ∑ k ( n k ) an−kbk etc. and others are related to the property of operator monotone functions. We also give a condition which implies the commutativity of a C*-algebra.