Modular Images Of Approximately Central Projections

It is shown that for any approximately central (AC) projection $e$ in the Flip orbifold $A_\theta^\Phi$ (of the irrational rotation C*-algebra $A_\theta$), and any modular automorphism $\alpha$ (arising from SL$(2,\mathbb Z)$), the AC projection $\alpha(e)$ is centrally Murray-von Neumann equivalent to one of the projections $e,\ \sigma(e),\ \kappa(e),\ \kappa^2(e),$ $\sigma\kappa(e),\ \sigma\kappa^2(e)$ in the $S_3$-orbit of $e,$ where $\sigma, \kappa$ are the Fourier and Cubic transforms of $A_\theta$. (The equivalence being implemented by an approximately central partial isometry in $A_\theta^\Phi$.) For smooth automorphisms $\alpha,\beta$ of the Flip orbifold $A_\theta^\Phi$, it is also shown that if $\alpha_*=\beta_*$ on $K_0(A_\theta^\Phi),$ then $\alpha(e)$ and $\beta(e)$ are centrally equivalent for each AC projection $e$.