Commutators in C*-algebras and traces

Dimension functions and traces on $C^\ast$-algebras are fundamental tools in the operator theory and its applications. A $C^*$-algebra is a complex Banach $\ast$-algebra $\mathcal{A}$ such that $\|A^*A\|=\|A\|^2$ for all $A \in \mathcal{A}$. For a $C^*$-algebra $\mathcal{A}$ by $\mathcal{A}^{\mathrm{id}}$ and $\mathcal{A}^+$ we denote its subsets of idempotents ($A=A^2$) and positive elements, respectively. If $A\in {\mathcal{A}}$, then $|A|=\sqrt{A^*A} \in \mathcal{A}^+$. An element $X\in \mathcal{A}$ is a commutator, if $X=[A, B]=AB-BA$ for some $A, B\in \mathcal{A}$. A mapping $\varphi : {\mathcal{A}}^+ \to [0,+\infty]$ is called a trace on a $C^*$-algebra $\mathcal{A}$, if $\varphi (X+Y)=\varphi(X)+\varphi(Y), \; \; \; \varphi (\lambda X)=\lambda \varphi (X)$ for all $ X,Y \in {\mathcal{A}}^+, \; \lambda \ge 0$ (moreover, $ 0\cdot(+\infty)\equiv 0$); $\varphi (Z^*Z)=\varphi (ZZ^*)$ for all $Z \in \mathcal{A}$. The following results were obtained. Let $\varphi$ be a faithful trace on a $C^*$-algebra $\mathcal{A}$; let $A,B \in \mathcal{A}^{\mathrm{id}}\setminus\{0\}$ be such that $ABA=\lambda A$ and $BAB=\lambda B$ for some $\lambda \in \mathbb{C}\setminus\{0, 1\}$. Then $[A, B]^n\not=0$ for all $n\in \mathbb{N}$. Corollary: Let $\varphi$ be a faithful tracial state on a $C^*$-algebra $\mathcal{A}$, let $A,B \in \mathcal{A}^{\mathrm{id}}\setminus\{0\}$ be such that $ABA=\lambda A$ and $BAB=\lambda B$ for some $\lambda \in \mathbb{C}\setminus\{0, 1\}$. Then the element $[A, B]^{2n}$ is a non-commutator for all $n\in \mathbb{N}$. Let $\mathcal{H}$ be a separable 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 separable 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. An operator $AP$ is a commutator if and only if $PA$ is a commutator.