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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Free states of the gauge invariant canonical anticommutation relations
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by B. M. Baker PDF
Trans. Amer. Math. Soc. 237 (1978), 35-61 Request permission

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
  • © Copyright 1978 American Mathematical Society
  • 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