Modular Images Of Approximately Central Projections

It is shown that for any approximately central (AC) projection $e$ in the Flip orbifold $A_{\theta }^{\mathrm{\Phi }}$ (of the irrational rotation C*-algebra $A_{\theta }$), and any modular automorphism $\alpha$ (arising from SL$(2,\mathbb Z)$),theACprojection$\alpha(e)$ 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 $\sigma, \kappa$aretheFourierandCubictransformsof$A_\theta$. (The equivalence being implemented by an approximately central partial isometry in $A_{\theta }^{\mathrm{\Phi }}$.) For smooth automorphisms $\alpha ,\beta$ of the Flip orbifold $A_{\theta }^{\mathrm{\Phi }}$, it is also shown that if ${\alpha }_{\ast }={\beta }_{\ast }$ on $K_0(A_{\theta }^{\mathrm{\Phi }}),$ then $\alpha (e)$ and $\beta (e)$ are centrally equivalent for each AC projection $e$.