Free states of the gauge invariant canonical anticommutation relations. II
Author:
B. M. Baker
Journal:
Trans. Amer. Math. Soc. 254 (1979), 135155
MSC:
Primary 81D05; Secondary 46L10
MathSciNet review:
539911
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Abstract: A class of representations of the gauge invariant subalgebra of the canonical anticommutation relations (henceforth GICAR) is studied. These representations are induced by restricting the wellknown pure, nongauge invariant generalized free states of the canonical anticommutation relations (henceforth CAR). Denoting a state of the CAR by , and the unique generalized free state of the CAR such that and by , it is shown that a pure, nongauge invariant state induces a factor representation of the GICAR if and only if .
 [1]
Huzihiro
Araki, On quasifree states of 𝐶𝐴𝑅 and
Bogoliubov automorphisms, Publ. Res. Inst. Math. Sci.
6 (1970/71), 385–442. MR 0295702
(45 #4768)
 [2]
B.
M. Baker, Free states of the gauge invariant
canonical anticommutation relations, Trans.
Amer. Math. Soc. 237 (1978), 35–61. MR 479361
(80b:46081), http://dx.doi.org/10.1090/S00029947197804793616
 [3]
E.
Balslev, J.
Manuceau, and A.
Verbeure, Representations of anticommutation relations and
Bogolioubov transformations, Comm. Math. Phys. 8
(1968), 315–326. MR 0253646
(40 #6860)
 [4]
Ola
Bratteli, Inductive limits of finite dimensional
𝐶*algebras, Trans. Amer. Math.
Soc. 171 (1972),
195–234. MR 0312282
(47 #844), http://dx.doi.org/10.1090/S00029947197203122822
 [5]
James
G. Glimm, On a certain class of operator
algebras, Trans. Amer. Math. Soc. 95 (1960), 318–340. MR 0112057
(22 #2915), http://dx.doi.org/10.1090/S00029947196001120575
 [6]
R.
Haag, The mathematical structure of the BardeenCooper Schrieffer
model., Nuovo Cimento (10) 25 (1962), 287–299
(English, with Italian summary). MR 0145921
(26 #3449)
 [7]
R. T. Powers, Thesis, Princeton University, Princeton, N. J., 1967.
 [8]
Robert
T. Powers, Representations of uniformly hyperfinite algebras and
their associated von Neumann rings, Ann. of Math. (2)
86 (1967), 138–171. MR 0218905
(36 #1989)
 [9]
Robert
T. Powers and Erling
Størmer, Free states of the canonical anticommutation
relations, Comm. Math. Phys. 16 (1970), 1–33.
MR
0269230 (42 #4126)
 [10]
G. Stamatopoulos, Thesis, University of Pennsylvania, Philadelphia, Pa., 1974.
 [11]
A.
van Daele and A.
Verbeure, Unitary equivalence of Fock representations on the Weyl
algebra, Comm. Math. Phys. 20 (1971), 268–278.
MR
0286406 (44 #3619)
 [1]
 H. Araki, On quasifree states of CAR and Bogoliubov automorphisms, Publ. Res. Inst. Math. Sci. Ser. A 6 (1970), 385442. MR 0295702 (45:4768)
 [2]
 B. M. Baker, Free states of the gauge invariant canonical anticommutation relations, Trans. Amer. Math. Soc. 237 (1978), 3561. MR 479361 (80b:46081)
 [3]
 E. Balslev, J. Manuceau and A. Verbeure, Representations of anticommutation relations and Bogoliubov transformations, Comm. Math. Phys. 8 (1968), 315326. MR 0253646 (40:6860)
 [4]
 O. Bratteli, Inductive limits of finite dimensional algebras, Trans. Amer. Math. Soc. 171 (1972), 195234. MR 0312282 (47:844)
 [5]
 J. Glimm, On a certain class of operator algebras, Trans. Amer. Math. Soc. 95 (1960), 318340. MR 0112057 (22:2915)
 [6]
 R. Haag, The mathematical structure of the BardeenCooperSchrieffer model, Nuovo Cimento 25 (1962), 287299. MR 0145921 (26:3449)
 [7]
 R. T. Powers, Thesis, Princeton University, Princeton, N. J., 1967.
 [8]
 , Representations of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. of Math. 86 (1967), 138171. MR 0218905 (36:1989)
 [9]
 R. T. Powers and E. Stormer, Free states of the canonical anticommutation relations, Comm. Math. Phys. 16 (1970), 133. MR 0269230 (42:4126)
 [10]
 G. Stamatopoulos, Thesis, University of Pennsylvania, Philadelphia, Pa., 1974.
 [11]
 A. Van Daele and A. Verbeure, Unitary equivalence of Fock representations on the Weyl algebra, Comm. Math. Phys. 20 (1971), 268278. MR 0286406 (44:3619)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197905399119
PII:
S 00029947(1979)05399119
Keywords:
Anticommutation relations,
gauge invariance,
approximately finite algebra,
generalized free states factor representations
Article copyright:
© Copyright 1979 American Mathematical Society
