Free States of the Gauge Invariant Canonical Anticommutation Relations

The gauge invariant subalgebra of the canonical anticommutation relations (henceforth GICAR) is viewed as an inductive limit of finitedimensional ${C^\ast }$-algebras, and a study is made of a simple class of its representations. In particular, representations induced by restricting the wellknown gauge invariant generalized free states from the entire canonical anticommutation relations (henceforth CAR) are considered. Denoting (a) a state of the CAR by $\omega$ and its restriction to the GICAR by ${\omega ^ \circ }$, (b) the unique gauge invariant generalized free state of the CAR such that $\omega (a{(f)^\ast }a(g)) = (f,Ag)$ by ${\omega _A}$, it is shown that $(1)\;\omega _A^ \circ$ induces (an impure) factor representation of the GICAR if and only if ${\text {Tr}}\;A(I - A) = \infty$, (2) two (impure) GICAR factor representations $\omega _A^ \circ$ and $\omega _B^\circ$ are quasi-equivalent if and only if ${A^{1/2}} - {B^{1/2}}$ and ${(I - A)^{1/2}} - {(I - B)^{1/2}}$ are Hilbert-Schmidt class operators.