K-Theory of Approximately Central Projections in the Flip Orbifold

For an approximately central (AC) Powers-Rieffel projection $e$ in the irrational Flip orbifold C*-algebra $A_\theta^\Phi,$ where $\Phi$ is the Flip automorphism of the rotation C*-algebra $A_\theta,$ we compute the Connes-Chern character of the cutdown of any projection by $e$ in terms of K-theoretic invariants of these projections. This result is then applied to computing a complete K-theoretic invariant for the projection $e$ with respect to central equivalence (within the orbifold). Thus, in addition to the canonical trace, there is a $4\times6$ K-matrix invariant $K(e)$ arising from unbounded traces of the cutdowns of a canonically constructed basis for $K_0(A_\theta^\Phi) = \mathbb Z^6$. Thanks to a theorem of Kishimoto, this enables us to tell when AC projections in $A_\theta^\Phi$ are Murray-von Neumann equivalent via an approximately central partial isometry (or unitary) in $A_\theta^\Phi$. As additional application, we obtain the K-matrix of canonical SL$(2,\mathbb Z)$-automorphisms of $e$ and show that there is a subsequence of $e$ such that $e, \sigma(e), \kappa(e), \kappa^2(e), \sigma\kappa(e), \sigma\kappa^2(e)$ -- which are the orbit elements of $e$ under the symmetric group $S_3 \subset$ SL$(2,\mathbb Z)$ -- are pairwise centrally not equivalent, and that each SL$(2,\mathbb Z)$ image of $e$ is centrally equivalent to one of these, where $\sigma, \kappa$ are the Fourier and Cubic transform automorphisms of the rotation algebra.