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 $0<varepsilonleq 1/2$, there exist $D,X in mathcal{B}(mathcal{H})$ with $|[D,X]-1_{mathcal{B}(mathcal{H})}|leq varepsilon$ such that $|D||X|=Oleft(log^5frac{1}{varepsilon}right)$, where $[D,X]:= DX-XD$. In this paper, we show that Taos result still holds for certain class of unital C*-algebras which include $mathcal{B}(mathcal{H})$ as well as the Cuntz algebra $mathcal{O}_2$.