Commutators Close to the Identity in Unital C*-Algebras

Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{B}(\mathcal{H})$ be the C*-algebra of all bounded linear operators on $\mathcal{H}$, equipped with the operator-norm. By improving the Brown-Pearcy construction, Terence Tao in 2018, extended the result of Popa [1981] which reads as : For each $0<\varepsilon\leq 1/2$, there exist $D,X \in \mathcal{B}(\mathcal{H})$ with $\|[D,X]-1_{\mathcal{B}(\mathcal{H})}\|\leq \varepsilon$ such that $\|D\|\|X\|=O\left(\log^5\frac{1}{\varepsilon}\right)$, where $[D,X]:= DX-XD$. In this paper, we show that Tao's result still holds for certain class of unital C*-algebras which include $\mathcal{B}(\mathcal{H})$ as well as the Cuntz algebra $\mathcal{O}_2$.