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 $0<\epsilon \le 1\mathrm{/}2$, there exist $D,X \in $\mathcal{B}$($\mathcal{H}$)$with$|[D,X]-1_{$\mathcal{B}$($\mathcal{H}$)}|\leq \varepsilon$suchthat$|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 ${\mathcal{O}}_2$.