down sampling and a factor of 2 without loss of information
If you down-sample without filtering, you will fold spectrum above half the new sample rate together with spectrum below half the new sample rate.  If both those spectral bands (above and below) are sparse enough (big gaps in the spectral content) in the right places, then it is possible that when folded, there will be no overlap of the folded non-zero high band spectrum with the non-zero low band spectrum, thus no loss of information due to aliasing.  If you know where the spectral gaps were, you could unfold the data (with some effort).

In the more typical case where those two spectrum bands (upper and lower) are not sparse enough and thus do overlap, the two spectrums be mixed (like mixing paint) when downsampling, thus incur an informationally lossy process.