Commutators close to the identity in unital C*-algebras

Abstract

Let H be an infinite dimensional Hilbert space and B(H) be the C∗-algebra of all bounded linear operators on H, equipped with the operator-norm. By improving the Brown–Pearcy construction, Tao (J. Oper. Theory82(2) (2019) 369–382) extended the result of Popa (On commutators in properly infinite W∗-algebras, in: Invariant subspaces and other topics (1982) (Boston, Mass.: Birkhäuser, Basel)) which reads as: for each 0 < ε≤ 1 / 2 , there exist D, X∈ B(H) with ‖ [D, X] - 1 <inf>B</inf><inf>(</inf><inf>H</inf><inf>)</inf>‖ ≤ ε such that ‖D‖‖X‖=O(log51ε), 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 B(H) as well as the Cuntz algebra O<inf>2</inf>. © 2022, Indian Academy of Sciences.