Abstract
Let τ be an action of a compact abelian groupG on aC*-algebraA, and assume that the fixed-point subalgebraA τ is an AF-algebra. We show that if δ is a closed *-derivation onA commuting with τ, and the restriction of δ toA τ generates a one-parameter group of *-automorphisms, then δ itself is a generator. In particular, the result applies if τ is an infinite product action ofG on a UHF algebra. Furthermore, if in this situation δ1 and δ2 are two derivations both satisfying the hypotheses on δ, and δ1 and δ2 have the same restriction toA τ, then there exists a one-parameter subgroup of the action τ with generator δ0 such thatD(δ1)∩D(δ2)∩D(δ0) is a joint core for the three derivations, and δ2=δ1+δ0 on this core.
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References
Aizenman, M.: On vector fields as generators of flows: A counterexample to Nelsons conjecture. Ann. Math.107, 287–296 (1978)
Araki, H., Haag, R., Kastler, D., Takesaki, M.: Extension of KMS states and chemical potential. Commun. Math. Phys.53, 97–134 (1977)
Batty, C. J. K.: Unbounded derivations of commutativeC*-algebras. Commun. Math. Phys.61, 419–425 (1978)
Bratteli, O., Jørgensen, P. E. T.: Unbounded derivations tangential to compact groups of automorphisms. J. Funct. Anal. (to appear)
Bratteli, O.: Inductive limits of finite-dimensionalC*-algebras. Trans. Am. Math. Soc.171, 195–234 (1972)
Bratteli, O.: Crossed products of UHF-algebras by product type actions. Duke Math. J.46, 1–23 (1979)
Bratteli, O., Robinson, D. W.: Operator algebras and quantum statistical mechanics I. New York, Heidelberg, Berlin: Springer 1979
Bratteli, O., Robinson, D. W.: Operator algebras and quantum statistical mechanics II. New York, Heidelberg, Berlin: Springer 1981
Bratteli, O., Kishimoto, A.: Generation of semigroups and two-dimensional quantum lattice systems. J. Funct. Anal.35, 344–368 (1980)
Bratteli, O., Evans, D. E.: Dynamical semigroups commuting with compact abelian actions. Warwick preprint (1982)
Brown, L. G.: Stable isomorphism of hereditary subalgebras ofC*-algebras. Pac. J. Math.71, 335–348 (1977)
Connes, A.: Une classification des facteurs de type III. Ann. Sci. Ecole Norm. Sup.6, 133–252 (1973)
Handelman, D., Rossmann, W.: Product type actions of finite and compact groups. Preprint
Kaplansky, I.: Modules over operator algebras. Am. J. Math.75, 839–859 (1953)
Kishimoto, A., Robinson, D. W.: On unbounded derivations commuting with a compact group of *-automorphisms. New South Wales preprint (1981)
Kishimoto, A., Takai, H.: Some remarks onC*-dynamical systems with a compact abelian group. Publ. R.I.M.S. Kyoto Univ.14, 383–397 (1978)
Powers, R. T., Price, G.: Derivations vanishing onS(∞). Commun. Math. Phys.84, 439–447 (1982)
Powers, R. T.: A remark on the domain of an unbounded derivation of aC*-algebra. J. Funct. Anal.18, 85–95 (1975)
Rosenberg, J.: Appendix to O. Bratteli's paper on: Crossed products of UHF algebras. Duke Math. J.46, 25–26 (1979)
Sakai, S.: On one parameter groups of *-automorphisms on operator algebras and the corresponding unbounded derivations. Am. J. Math.98, 427–440 (1976)
Wassermann, A. J.: Automorphic actions of compact groups on operator algebras. University of Pennsylvania Thesis (1981)
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Communicated by H. Araki
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Bratteli, O., Jørgensen, P.E.T. Derivations commuting with abelian gauge actions on lattice systems. Commun.Math. Phys. 87, 353–364 (1982). https://doi.org/10.1007/BF01206028
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DOI: https://doi.org/10.1007/BF01206028