On The Even CAR Algebra

Abstract

We exhibit a family of *-isomorphisms mapping the CAR algebra onto its even subalgebra. In 1970 E. Stormer proved that the even subalgebra of the CAR algebra over an infinite dimensional separable Hilbert space is UHF of type 2 ∞ , hence *-isomorphic to the CAR algebra itself (1). But it seems to be unknown that such isomorphisms have a nice and sim- ple realization in terms of Bogoliubov endomorphisms with "statistical dimension" √ 2. Bogoliubov endomorphisms are conveniently described using Araki's "selfdual" CAR algebra formalism (2). Let K be an infinite dimensional separable complex Hilbert space, equipped with a complex conjugation k 7→ k ∗ , and let C(K) denote the unique (simple) C*-algebra generated by 1 and the elements of K, subject to the anticommutation relation k ∗ k ' + kk ∗ = h k, ki 1, k, k ' ∈ K.