Commutators Close to the Identity in Unital C*-Algebras

Abstract

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 $0<\epsilon \le 1/2$, there exist $D,X\in \mathcal{B}(\mathcal{H})$ with $\Vert [D,X]-1_{\mathcal{B}(\mathcal{H})}\Vert \le \epsilon$ such that $\Vert D\Vert \Vert X\Vert =O\left({\log }^5\frac{1}{\epsilon }\right)$, 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 ${\mathcal{O}}_2$.

10 pages