Stability of A\(\mathbb{T}\)-relations in \(C^*\)-algebras with tracial rank at most one

Abstract

An old and famous problem from the 1950s, popularized by Halmos, is that whether any pair of almost commuting contractive self-adjoint matrices are norm close to a pair of exactly commuting self-adjoint matrices? This question was solved affirmatively by Lin in the 1990's. In this paper, we study the general Halmos problem concerning unitary elements in \(C^*\)-algebras. Specifically, we first introduce the definition of A\(\mathbb{T}\)-relations, and then we give a necessary and sufficient condition for the stability of A\(\mathbb{T}\)-relations in any unital infinite dimensional simple separable \(C^*\)-algebra with tracial rank at most one. Finally, as applications, we show that many naturally occurring relations are A\(\mathbb{T}\)-relations, and thus the stability results of these relations can be obtained by applying the above conclusions.