In this talk we discuss the structure of C*-diagonals (in the sense of Kumjian) that appear in the CAR algebra. We present a construction of a free minimal action on the Cantor space based on the so-called paper-folding sequence, that gives rise to a Cantor spectrum diagonal that is not conjugate to a standard AF diagonal. This allows us to see that in fact the CAR algebra admits countably many pairwise non-conjugate Cantor spectrum diagonals, distinguished by their diagonal dimension (as introduced by Li, Liao and Winter in 2023). This is joint work with Wilhelm Winter.
