Modular Images Of Approximately Central Projections

Abstract

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