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

We show that if theta is an element of(0, 1) is an irrational number, then for any epsilon > 0, there exists delta > 0 satisfying the following: For any unital C*-algebra A with the cancellation property, strict comparison and nonempty tracial state space, any three unitaries u, v, w is an element of A such that (1) ||vu - e(2 pi i theta) uv|| < delta, w(3) = 1(A), wuw(-1) = e(-pi i theta) u(-1)v, wvw(-1) =u(-1), (2) tau(aw) = 0, tau(aw(2)) = 0 and tau((vuv*u*)(n)) = e(2 pi in theta) for all n is an element of N, all a is an element of C*(u, v) and all tracial state tau on A, where C*(u, v) is the C*-subalgebra generated by u and v, there exists a triple of unitaries <(u)over tilde>, (v) over tilde, (w) over tilde is an element of A such that (v) over tilde(u) over tilde = e(2 pi i theta) (u) over tilde(v) over tilde, (w) over tilde (3) = 1(A), (w) over tilde(u) over tilde(w) over tilde (-1) = e(-pi i theta) (u) over tilde (-1)(v) over tilde, (w) over tilde(v) over tilde(w) over tilde (-1) = (u) over tilde (-1), ||u - (u) over tilde|| < epsilon, ||v - <(v)over tilde>|| < epsilon, ||w - <(w)over tilde>|| < epsilon. According to the above conclusion, the rotation relation of two unitaries with Z(3)-action is stable under the conditions in (2). Likewise, we show that the rotation relation of two unitaries with Z(4) and Z(6)-actions is stable under the above similar conditions.