Perturbations of ground states of type algebras
Author:
C. J. K. Batty
Journal:
Proc. Amer. Math. Soc. 78 (1980), 539544
MSC:
Primary 46L30; Secondary 81C12, 82A15
MathSciNet review:
556628
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Abstract: It is shown that the class of irreducible representations of a type I algebra A which satisfy a spectrum condition for a given dynamical system on A is unchanged if the system undergoes a sufficiently small relatively bounded perturbation. It follows that if A is also unital, then the existence of ground states is unaffected by such perturbations.
 [1]
Huzihiro
Araki, Relative Hamiltonian for faithful normal states of a von
Neumann algebra, Publ. Res. Inst. Math. Sci. 9
(1973/74), 165–209. MR 0342080
(49 #6826)
 [2]
William
Arveson, On groups of automorphisms of operator algebras, J.
Functional Analysis 15 (1974), 217–243. MR 0348518
(50 #1016)
 [3]
C.
J. K. Batty, Small perturbations of 𝐶*dynamical
systems, Comm. Math. Phys. 68 (1979), no. 1,
39–43. MR
539735 (80h:46113)
 [4]
Ola
Bratteli and Derek
W. Robinson, Unbounded derivations of 𝐶*algebras,
Comm. Math. Phys. 42 (1975), 253–268. MR 0377526
(51 #13698)
 [5]
Ola
Bratteli and Derek
W. Robinson, Unbounded derivations of 𝐶*algebras. II,
Comm. Math. Phys. 46 (1976), no. 1, 11–30. MR 0390785
(52 #11608)
 [6]
Ola
Bratteli and Derek
W. Robinson, Operator algebras and quantumstatistical mechanics.
II, SpringerVerlag, New YorkBerlin, 1981. Equilibrium states. Models
in quantumstatistical mechanics; Texts and Monographs in Physics. MR 611508
(82k:82013)
 [7]
Tosio
Kato, Perturbation theory for linear operators, 2nd ed.,
SpringerVerlag, BerlinNew York, 1976. Grundlehren der Mathematischen
Wissenschaften, Band 132. MR 0407617
(53 #11389)
 [8]
Christopher
Lance and Assadollah
Niknam, Unbounded derivations of group
𝐶*algebras, Proc. Amer. Math. Soc.
61 (1976), no. 2,
310–314 (1977). MR 0428051
(55 #1080), http://dx.doi.org/10.1090/S00029939197604280518
 [9]
Roberto
Longo, Automatic relative boundedness of derivations in
𝐶*algebras, J. Funct. Anal. 34 (1979),
no. 1, 21–28. MR 551107
(81f:46072), http://dx.doi.org/10.1016/00221236(79)900223
 [10]
Dorte
Olesen, On spectral subspaces and their applications to
automorphism groups, Symposia Mathematica, Vol. XX (Convegno sulle
Algebre 𝐶* e loro Applicazioni in Fisica Teorica, Convegno sulla
Teoria degli Operatori Indice e Teoria 𝐾, INDAM, Rome, 1975)
Academic Press, London, 1976, pp. 253–296. MR 0487481
(58 #7110)
 [11]
Robert
T. Powers and Shôichirô
Sakai, Existence of ground states and KMS states for approximately
inner dynamics, Comm. Math. Phys. 39 (1974/75),
273–288. MR 0359623
(50 #12075)
 [12]
Michael
Reed and Barry
Simon, Methods of modern mathematical physics. I, 2nd ed.,
Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York,
1980. Functional analysis. MR 751959
(85e:46002)
 [13]
S. Sakai, The theory of unbounded derivations in algebras, Newcastle Univ. Lecture Notes, 1977.
 [1]
 H. Araki, Relative Hamiltonian for faithful normal states of a von Neumann algebra, Publ. Res. Inst. Math. Sci. 9 (1973), 165209. MR 0342080 (49:6826)
 [2]
 W. B. Arveson, On groups of automorphisms of operator algebras, J. Functional Analysis 15 (1974), 217243. MR 0348518 (50:1016)
 [3]
 C. J. K. Batty, Small perturbations of dynamical systems, Comm. Math. Phys. 68 (1978), 3943. MR 539735 (80h:46113)
 [4]
 O. Bratteli and D. W. Robinson, Unbounded derivations of algebras, Comm. Math. Phys. 42 (1975), 253268. MR 0377526 (51:13698)
 [5]
 , Unbounded derivations of algebras. II, Comm. Math. Phys. 46 (1976), 1130. MR 0390785 (52:11608)
 [6]
 , Operator algebras and quantum statistical mechanics. II, Springer, Berlin (to appear). MR 611508 (82k:82013)
 [7]
 T. Kato, Perturbation theory for linear operators, 2nd ed., Springer, Berlin, 1976. MR 0407617 (53:11389)
 [8]
 E. C. Lance and A. Niknam, Unbounded derivations of group algebras, Proc. Amer. Math. Soc. 6 (1976), 310314. MR 0428051 (55:1080)
 [9]
 R. Longo, Automatic relative boundedness of derivations in algebras, J. Functional Analysis 34 (1979), 2128. MR 551107 (81f:46072)
 [10]
 D. Olesen, On spectral subspaces and their applications to automorphism groups, Symposia Mathematics vol. 20, Academic Press, New York, 1976, pp. 253296. MR 0487481 (58:7110)
 [11]
 R. T. Powers and S. Sakai, Existence of ground states and KMS states for approximately inner dynamics, Comm. Math. Phys. 39 (1975), 273288. MR 0359623 (50:12075)
 [12]
 M. Reed and B. Simon, Methods of modern mathematical physics. II, Academic Press, New York, 1975. MR 751959 (85e:46002)
 [13]
 S. Sakai, The theory of unbounded derivations in algebras, Newcastle Univ. Lecture Notes, 1977.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198005566289
PII:
S 00029939(1980)05566289
Keywords:
Type I algebra,
small perturbation,
ground state,
spectrum condition
Article copyright:
© Copyright 1980
American Mathematical Society
