Extensions of representations of the CAR algebra to the Cuntz algebra O 2 — the Fock and the infinite wedge —

(1.1) anam + a ∗ man = δn,mI, a ∗ na ∗ m + a ∗ ma ∗ n = anam + aman = 0 for n, m ∈ N. A0 always has unique C∗-norm ‖ · ‖ and the completion A ≡ A0 with respect to ‖ · ‖ is called the CAR algebra in theory of operator algebras([6]). In [1, 2, 3, 4], we construct several polynomial embeddings of A into the Cuntz algebrasON . For example, if s1, s2 are canonical generators of O2, that is, they satisfy (1.2) si sj = δijI (i, j = 1, 2), s1s ∗ 1 + s2s ∗ 2 = I, then (1.3) a1 ≡ s1s2, an ≡ ∑