Paper-folding models for the CAR algebra

We show that the CAR algebra admits a Cantor spectrum C*-diagonal that is not conjugate to the standard AF diagonal. We obtain this by classification theory of C*-algebras, and the diagonal arises by realising the CAR algebra as the crossed product of a free minimal action on the Cantor space, where the acting group is the product of a locally finite group with the infinite dihedral group. The main ingredient in the construction is a binary subshift associated to the well-known regular paper-folding sequence. Moreover, we show that the CAR algebra in fact admits countably many, pairwise non-conjugate, Cantor spectrum diagonals which are distinguished by the different values of their diagonal dimension, as defined by Li, Liao and the second named author.