Perturbations of ground states of type 𝐼 𝐶*-algebras

Batty, C. J. K.

doi:10.1090/S0002-9939-1980-0556628-9

Perturbations of ground states of type $\textrm {I}$ $C^{\ast }$-algebras
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by C. J. K. Batty PDF

Proc. Amer. Math. Soc. 78 (1980), 539-544 Request permission

Abstract:

It is shown that the class of irreducible representations of a type I ${C^\ast }$-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.

References

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, DOI 10.2977/prims/1195192744
William Arveson, On groups of automorphisms of operator algebras, J. Functional Analysis 15 (1974), 217–243. MR 0348518, DOI 10.1016/0022-1236(74)90034-2
C. J. K. Batty, Small perturbations of $C^{\ast }$-dynamical systems, Comm. Math. Phys. 68 (1979), no. 1, 39–43. MR 539735, DOI 10.1007/BF01562540
Ola Bratteli and Derek W. Robinson, Unbounded derivations of $C^{\ast }$-algebras, Comm. Math. Phys. 42 (1975), 253–268. MR 377526, DOI 10.1007/BF01608976
Ola Bratteli and Derek W. Robinson, Unbounded derivations of $C^{\ast }$-algebras. II, Comm. Math. Phys. 46 (1976), no. 1, 11–30. MR 390785, DOI 10.1007/BF01610496
Ola Bratteli and Derek W. Robinson, Operator algebras and quantum-statistical mechanics. II, Texts and Monographs in Physics, Springer-Verlag, New York-Berlin, 1981. Equilibrium states. Models in quantum-statistical mechanics. MR 611508, DOI 10.1007/978-3-662-09089-3
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
Christopher Lance and Assadollah Niknam, Unbounded derivations of group $C^*$-algebras, Proc. Amer. Math. Soc. 61 (1976), no. 2, 310–314 (1977). MR 428051, DOI 10.1090/S0002-9939-1976-0428051-8
Roberto Longo, Automatic relative boundedness of derivations in $C^{\ast }$-algebras, J. Functional Analysis 34 (1979), no. 1, 21–28. MR 551107, DOI 10.1016/0022-1236(79)90022-3
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
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 359623, DOI 10.1007/BF01705375
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

The theory of unbounded derivations in

algebras