Paper-folding models for the CAR algebra

TL;DR

This paper constructs new types of diagonals in the CAR algebra using dynamical systems and classification theory, revealing a rich structure of non-conjugate diagonals with diverse properties.

Contribution

It introduces novel Cantor spectrum diagonals in the CAR algebra that are not conjugate to the standard AF diagonal, expanding understanding of its internal structure.

Findings

Existence of non-conjugate Cantor spectrum diagonals in the CAR algebra.

Construction of diagonals via crossed products with free minimal actions.

Identification of diagonals distinguished by their diagonal dimension.

Abstract

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.