Recursive Fermion System in Cuntz Algebra.I -- Embeddings of Fermion Algebra into Cuntz Algebra ---

Embeddings of the CAR (canonical anticommutation relations) algebra of fermions into the Cuntz algebra ${\cal O}_2$ (or ${\cal O}_{2d}$ more generally) are presented by using recursive constructions. As a typical example, an embedding of CAR onto the U(1)-invariant subalgebra of ${\cal O}_2$ is constructed explicitly. Generalizing this construction to the case of ${\cal O}_{2^p}$, an embedding of CAR onto the U(1)-invariant subalgebra of ${\cal O}_{2^p}$ is obtained. Restricting a permutation representation of the Cuntz algebra, we obtain the Fock representation of CAR. We apply the results to embed the algebra of parafermions of order $p$ into ${\cal O}_{2^p}$ according to the Green's ansatz.