  Previously, we have shown that the CAR algebra for fermions is embedded in
the Cuntz algebra ${\cal O}_{2}$ in such a way that the generators are
expressed in terms of polynomials in the canonical generators of the latter,
and it coincides with the U(1)-fixed point subalgebra ${\cal A}\equiv {\cal
O}_{2}^{U(1)}$ of ${\cal O}_{2}$ for the canonical gauge action. Based on this
embedding formula, some properties of ${\cal A}$ are studied in detail by
restricting those of ${\cal O}_{2}$. Various endomorphisms of ${\cal O}_{2}$,
which are defined by polynomials in the canonical generators, are explicitly
constructed, and transcribed into those of ${\cal A}$. Especially, we
investigate branching laws for a certain family of such endomorphisms with
respect to four important representations, i.e., the Fock representation, the
infinite wedge representation and their duals. These endomorphisms are
completely classified by their branching laws. As an application, we show that
the reinterpretation of the Fock vacuum as the Dirac vacuum is described in
representation theory through a mixture of fermions.
