Recursive Fermion System in Cuntz Algebra . II — Endomorphism , Automorphism and Branching of Representation — Mitsuo Abe

Based on an embedding formula of the CAR algebra into the Cuntz algebra O2p , properties of the CAR algebra are studied in detail by restricting those of the Cuntz algebra. Various ∗-endomorphisms of the Cuntz algebra are explicitly constructed, and transcribed into those of the CAR algebra. In particular, a set of ∗-endomorphisms of the CAR algebra into its even subalgebra are constructed. According to branching formulae, which are obtained by composing representations and ∗-endomorphisms, it is shown that a KMS state of the CAR algebra is obtained through the above even-CAR endomorphisms from the Fock representation. A U(2) action onO2p induces ∗-automorphisms of the CAR algebra, which are given by nonlinear transformations expressed in terms of polynomials in generators. It is shown that, among such ∗-automorphisms of the CAR algebra, there exists a family of one-parameter groups of ∗-automorphisms describing time evolutions of fermions, in which the particle number of the system changes by time while the Fock vacuum is kept invariant. E-mail address: [email protected] E-mail address: [email protected] – 1 – §