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 $A_θ^Φ,$ where $Φ$ is the Flip automorphism of the rotation C*-algebra $A_θ,$ 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 $K_0(A_θ^Φ) = \mathbb Z^6$. Thanks to a theorem of Kishimoto, this enables us to tell when AC projections in $A_θ^Φ$ are Murray-von Neumann equivalent via an approximately central partial isometry (or unitary) in $A_θ^Φ$. 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 $e, σ(e), κ(e), κ^2(e), σκ(e), σκ^2(e)$ -- 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 $σ, κ$ are the Fourier and Cubic transform automorphisms of the rotation algebra.