Paper-folding models for the CAR algebra

<jats:p> We show that the canonical anticommutation relations (CAR) algebra admits a Cantor spectrum <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> <jats:tex-math>$\mathrm {C}^\ast $</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S0143385726103125_inline1.png"> <jats:alt-text content-type="machine-generated">normal upper C Superscript asterisk</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> -diagonal that is not conjugate to the standard AF diagonal. We obtain this by classification theory of <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> <jats:tex-math>$\mathrm {C}^\ast $</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S0143385726103125_inline2.png"> <jats:alt-text content-type="machine-generated">normal upper C Superscript asterisk</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> -algebras, and the diagonal arises by realizing 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. </jats:p>