# Symmetries of the CAR algebra

### Abstract

A ${}^\ast$-automorphism of period two of the UHF $C^\ast$-algebra of type $2^\infty$ is constructed for which the fixed-point algebra is not approximately finite dimensional. The construction also yields a diagonal maximal commutative $C^\ast$-subalgebra whose spectrum is not totally disconnected. Variations and consequences of the construction are discussed.