Modular Images Of Approximately Central Projections

TL;DR

This paper classifies approximately central projections in the Flip orbifold of an irrational rotation algebra under modular automorphisms, showing their equivalence classes form an orbit under certain transformations and linking automorphisms via K-theory.

Contribution

It provides a detailed classification of AC projections under modular automorphisms in the Flip orbifold, extending understanding of their equivalence and automorphism actions.

Findings

AC projections are classified up to Murray-von Neumann equivalence within an orbit of transformations.

Automorphisms with the same K_0 action produce centrally equivalent projections.

The classification involves Fourier and Cubic transforms forming an S_3-orbit.

Abstract

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 $e$.