Commutators in C*-algebras and traces

Let H be a Hilbert space, dimH = +infinity. Let X = U |X| be the polar decomposition of an operator X is an element of B(H). Then, X is a non-commutator if and only if both U and |X| are non-commutators. A Hermitian operator X is an element of B(H) is a commutator if and only if the Cayley transform K(X) is a commutator. Let H be a Hilbert space and dim H <= +infinity, A, B, P is an element of B(H) and P = P2. If AB = lambda B A for some lambda is an element of C\{1} then the operator AB is a commutator. The operator AP is a commutator if and only if P A is a commutator.