STABILITY OF ROTATION RELATION OF TWO UNITARIES WITH ℤ3, ℤ4 AND ℤ6-ACTIONS IN C*-ALGEBRAS

Abstract

We show that if $𝜃\in (0,1)$ is an irrational number, then for any , there exists satisfying the following: For any unital $C^{\ast }$-algebra $A$ with the cancellation property, strict comparison and nonempty tracial state space, any three unitaries $u,v,w\in A$ such that

• (1) , $w^3=1_A$, $wu{w}^{-1}=e^{-\pi i𝜃}{u}^{-1}v$, $wv{w}^{-1}=u^{-1}$,

• (2) $\tau (aw)=0$, $\tau (a{w}^2)=0$ and $\tau ({(vu{v}^{\ast }{u}^{\ast })}^n)=e^{2\pi in𝜃}$ for all $n\in \mathbb{N}$, all $a\in C^{\ast }(u,v)$ and all tracial state $\tau$ on $A$, where $C^{\ast }(u,v)$ is the $C^{\ast }$-subalgebra generated by $u$ and $v$,

there exists a triple of unitaries $\tilde{u},\tilde{v},\tilde{w}\in A$ such that

According to the above conclusion, the rotation relation of two unitaries with ${\mathbb{Z}}_3$-action is stable under the conditions in (2). Likewise, we show that the rotation relation of two unitaries with ${\mathbb{Z}}_4$ and ${\mathbb{Z}}_6$-actions is stable under the above similar conditions.