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Free states of the gauge invariant canonical anticommutation relations


Author: B. M. Baker
Journal: Trans. Amer. Math. Soc. 237 (1978), 35-61
MSC: Primary 46L60; Secondary 81E05
DOI: https://doi.org/10.1090/S0002-9947-1978-0479361-6
MathSciNet review: 479361
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Abstract: 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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0479361-6
Keywords: Anticommutation relations, gauge invariance, approximately finite $ {C^\ast}$-algebra, generalized free states, factor representations, quasi-equivalent representations
Article copyright: © Copyright 1978 American Mathematical Society

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