Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 479361
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46L60, 81E05

Retrieve articles in all journals with MSC: 46L60, 81E05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0479361-6
PII: S 0002-9947(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