Commutators in $C*$-algebras and traces

Abstract

Let $\mathcal{H}$ be a Hilbert space, $\dim \mathcal{H}= +\infty$. Let $X=U|X|$ be the polar decomposition of an operator $X\in \mathcal{B}(\mathcal{H})$. Then $X$ is a non-commutator if and only if both $U$ and $|X|$ are non-commutators. A Hermitian operator $X\in \mathcal{B}(\mathcal{H})$ is a commutator if and only if the Cayley transform $\mathcal{K}(X)$ is a commutator. Let $\mathcal{H}$ be a Hilbert space and $\dim mathcal{H}\leq +\infty$, $A,B, P\in \mathcal{B}(\mathcal{H})$ and $P=P^2$. If $AB=\lambda BA$ for some $\lambda \in \mathbb{C}\setminus\{1\}$ then the operator $AB$ is a commutator. The operator $AP$ is a commutator if and only if $PA$ is a commutator.