In appendix C.4.3 of "Anyons in an exactly solved model and beyond", Kitaev provides a proof of the fact that when the Chern number is odd, a vortex in the gauge field accompanies an unpaired majorana mode.
The original gauge configuration that appears in the quadratic hamiltonian of majorana operators is B, while the one with a vortex is B'. He tries to prove the relative majorana number of them is $(-1)^{\nu(P)}$ where $P=\frac{1}{2}(1-iB)$ is the spectral projector and $\nu(P)$ is its Chern number. He defines the projection operator to x>0 plane and y>0 plane to be $\Pi^{(x)}$ and $\Pi^{(y)}$ respectively, and defines $X=i(P\Pi^{(x)}P+(1-P)\Pi^{(x)}(1-P))$. Then he express the relative majorana number as $$M(B,B')=\det(e^{-\pi X}e^{-i\pi\Pi^{(y)}} e^{\pi X}e^{i\pi\Pi^{(y)}})=\exp(i\pi^2Tr[X,\Pi^{(y)}])$$
$\Pi^y$ should be a diagonal matrix with $\Pi^y_{ii}=1$ if i corresponds to a site with $y>0$ and $\Pi^y_{ii}=0$ otherwise. Then $\Pi^y X$ and $X\Pi^y$ should have the same elements on the diagonal line, thus I can't see why the trace of their commutator is non-zero, even if they are infinite dimensional.
In the next line, while calculating $Tr[X,\Pi^{(y)}]$, he seems to claim $Tr(P^2[X,\Pi^{(y)}])=Tr(P[X,\Pi^{(y)}]P)$. I can't see why the cyclic property of trace applies here these are infinite dimensional matrices.
