Bulletin of the American Mathematical Society

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Book Information:

Author: Ola Bratteli
Title: Derivations, dissipations and group actions on $C^ *$-algebras
Additional book information: Lecture Notes in Mathematics, vol. 1229, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1986, vi + 277 pp., $23.60. ISBN 0-387-17199-1.

Charles A. Akemann and Gert K. Pedersen, Central sequences and inner derivations of separable $C^{\ast }$-algebras, Amer. J. Math. 101 (1979), no. 5, 1047–1061. MR 546302, DOI 10.2307/2374125

B. Blackadar, Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006. Theory of $C^*$-algebras and von Neumann algebras; Operator Algebras and Non-commutative Geometry, III. MR 2188261, DOI 10.1007/3-540-28517-2

Ola Bratteli, George A. Elliott, and Palle E. T. Jorgensen, Decomposition of unbounded derivations into invariant and approximately inner parts, J. Reine Angew. Math. 346 (1984), 166–193. MR 727402

4.

O. Bratteli, G. A. Elliott, F. M. Goodman and P. E. T. Jorgensen, Smooth Lie group actions on non-commutative tori, Preprint, 1986.

5a. O. Bratteli and D. W. Robinson, Unbounded derivations of C*-algebras, Comm. Math. Phys. 42 (1975), 253-268. MR 377526

5b. O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics. I, II, Springer-Verlag, New York, 1979/81 (vol. I in new ed., 1987).

6a. A. Connes, C*-algèbres et géométrie différentielle, C. R. Acad. Sci. Paris 290 (1980), 599-604. MR 572645

6b. A. Connes, Noncommutative differential geometry. I, The Chern character in K-homology; II, deRham homology and noncommutative algebra, Publ. Math. IHES 62 (1985). MR 823176

Ola Bratteli and Derek W. Robinson, Unbounded derivations of $C^{\ast }$-algebras, Comm. Math. Phys. 42 (1975), 253–268. MR 377526

8.

G. A. Elliott, Some C*-algebras with outer derivations. III, Ann. of Math. (2) 106 (1977), 121-143. MR 448093

Alain Connes, $C^{\ast }$ algèbres et géométrie différentielle, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 13, A599–A604 (French, with English summary). MR 572645

Alain Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985), 257–360. MR 823176

14.

T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1976. MR 407617

George A. Elliott, Some $C^*$-algebras with outer derivations. III, Ann. of Math. (2) 106 (1977), no. 1, 121–143. MR 448093, DOI 10.2307/1971162

R. Haag, N. M. Hugenholtz, and M. Winnink, On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys. 5 (1967), 215–236. MR 219283

Palle E. T. Jorgensen and Robert T. Moore, Operator commutation relations, Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht, 1984. Commutation relations for operators, semigroups, and resolvents with applications to mathematical physics and representations of Lie groups. MR 746138, DOI 10.1007/978-94-009-6328-3

Paul S. Muhly (ed.), Operator algebras and mathematical physics, Contemporary Mathematics, vol. 62, American Mathematical Society, Providence, RI, 1987. MR 878371, DOI 10.1090/conm/062

Richard V. Kadison, Transformations of states in operator theory and dynamics, Topology 3 (1965), no. suppl, suppl. 2, 177–198. MR 169073, DOI 10.1016/0040-9383(65)90075-3

Richard V. Kadison, Operator algebras—the first forty years, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 1–18. MR 679692

Richard V. Kadison and John R. Ringrose, Cohomology of operator algebras. I. Type $I$ von Neumann algebras, Acta Math. 126 (1971), 227–243. MR 283578, DOI 10.1007/BF02392032

Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR 719020

Irving Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839–858. MR 58137, DOI 10.2307/2372552

Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617

Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104

Robert T. Powers, Resistance inequalities for KMS states of the isotropic Heisenberg model, Comm. Math. Phys. 51 (1976), no. 2, 151–156. MR 426743

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

Marc A. Rieffel, $C^{\ast }$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415–429. MR 623572

Shôichirô Sakai, On a conjecture of Kaplansky, Tohoku Math. J. (2) 12 (1960), 31–33. MR 112055, DOI 10.2748/tmj/1178244484

19b. S. Sakai, The theory of unbounded derivatives in C*-algebras, Lecture Notes, Univ. of Newcastle, England, 1977.

Irving Segal, Notes towards the construction of nonlinear relativistic quantum fields. III. Properties of the $C^{\ast }$-dynamics for a certain class of interactions, Bull. Amer. Math. Soc. 75 (1969), 1390–1395. MR 251992, DOI 10.1090/S0002-9904-1969-12429-8

Ja. G. Sinaĭ and A. Ja. Helemskiĭ, A description of differentiations in algebras the type of algebras of local observables of spin systems, Funkcional. Anal. i Priložen. 6 (1972), no. 4, 99–100 (Russian). MR 0317686

Hiroshi Takai, On a problem of Sakai in unbounded derivations, J. Functional Analysis 43 (1981), no. 2, 202–208. MR 633976, DOI 10.1016/0022-1236(81)90029-X

C. T. Yang, Hilbert’s fifth problem and related problems on transformation groups, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Proc. Sympos. Pure Math., Vol. XXVIII, Amer. Math. Soc., Providence, R.I., 1976, pp. 142–146. MR 0425999

1.

C. A. Akemann and G. K. Pederson, Central sequences and inner derivations of separable C*-algebras, Amer. J. Math. 101 (1979), 1047-1061. MR 0546302

2.

B. Blackadar, K-theory for operator algebras, Springer-Verlag, New York, 1986. MR 2188261

3.

O. Bratteli, G. A. Elliott and P. E. T. Jorgensen, Decomposition of unbounded derivations into invariant and approximately inner parts, J. Reine Angew. Math. 346 (1984), 166-193. MR 727402

4.

O. Bratteli, G. A. Elliott, F. M. Goodman and P. E. T. Jorgensen, Smooth Lie group actions on non-commutative tori, Preprint, 1986.

5a. O. Bratteli and D. W. Robinson, Unbounded derivations of C*-algebras, Comm. Math. Phys. 42 (1975), 253-268. MR 377526




5b. O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics. I, II, Springer-Verlag, New York, 1979/81 (vol. I in new ed., 1987).




6a. A. Connes, C*-algèbres et géométrie différentielle, C. R. Acad. Sci. Paris 290 (1980), 599-604. MR 572645




6b. A. Connes, Noncommutative differential geometry. I, The Chern character in K-homology; II, deRham homology and noncommutative algebra, Publ. Math. IHES 62 (1985). MR 823176




7.

J. Dixmier, Les algèbres d'opérateurs dans l'espace Hilbertien, 2nd ed., Gauthier-Villars, Paris, 1969.

8.

G. A. Elliott, Some C*-algebras with outer derivations. III, Ann. of Math. (2) 106 (1977), 121-143. MR 448093

9.

R. Haag, N. M. Hugenholtz and M. Winnink, On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys. 5 (1967), 215-236. MR 219283

10a. P. E. T. Jorgensen and R. T. Moore, Operator commutation relations, D. Reidel Publishing Co., Dordrecht-Boston, 1984. MR 746138




10b. P. E. T. Jorgensen and P. S. Muhly (eds. ), Operator algebras and mathematical physics, Contemp. Math., vol. 62, Amer. Math. Soc., Providence, R. I., 1987. MR 878371




11a. R. V. Kadison, Transformation of states in operator theory and dynamics, Topology 3 (1965), 177-198. MR 169073




11b. R. V. Kadison (ed. ), Operator algebras and applications, Proc. Sympos. Pure Math., vol. 38, parts 1 and 2, Amer. Math. Soc., Providence, R. I., 1982. MR 679692




12a. R. V. Kadison and J. R. Ringrose, Cohomology of operator algebras. I, Type I von Neumann algebras, Acta Math. 126 (1971), 224-244. MR 283578




12b. R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras, vols. I and II, Academic Press, 1983/86. MR 719020




13.

I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839-859. MR 58137

14.

T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1976. MR 407617

15.

D. Montgomery and L. Zippin, Topological transformation groups, Interscience Publishers, Inc., New York, 1955. MR 73104

16.

R. T. Powers, Resistance inequalities for KMS-states of the isotropic Heisenberg model, Comm. Math. Phys. 51 (1976), 151-156. MR 426743

17.

R. T. Powers and S. Sakai, Existence of ground states and KMS states for approximately inner dynamics, Comm. Math. Phys. 39 (1975), 273-288. MR 359623

18.

M. A. Rieffel, C*-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415-429. MR 623572

19a. S. Sakai, On a conjecture of Kaplansky, Tôhoku Math. J. 12 (1960), 31-33. MR 112055




19b. S. Sakai, The theory of unbounded derivatives in C*-algebras, Lecture Notes, Univ. of Newcastle, England, 1977.




20.

I. E. Segal, Notes towards the construction of nonlinear relativistic quantum fields. III, Properties of the C*-dynamics for a certain class of interactions, Bull. Amer. Math. Soc. 75 (1969), 1390-1395. MR 251992

21.

Ja. G. Sinaĭ and A. Ja. Helmskiĭ, A description of differentiations in algebras of the type of local observables of spin systems, Funktsional Anal. i Prilozhen 6 (1972), 99-100; English transl., Funct. Anal. Appl. 6 (1973), 343-344. MR 317686

22.

H. Takai, On a problem of Sakai in unbounded derivations, J. Funct. Anal. 43 (1981), 202-208. MR 633976

23.

C. -T. Yang, Hilbert's fifth problem and related problems on transformation groups, Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, R. I., 1976, pp. 142-146. MR 425999

13. I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839-859. 14. T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1976. 15. D. Montgomery and L. Zippin, Topological transformation groups, Interscience Publishers, Inc., New York, 1955. 16. R. T. Powers, Resistance inequalities for KMS-states of the isotropic Heisenberg model, Comm. Math. Phys. 51 (1976), 151-156. 17. R. T. Powers and S. Sakai, Existence of ground states and KMS states for approximately inner dynamics, Comm. Math. Phys. 39 (1975), 273-288. 18. M. A. Rieffel, C*-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415-429. 19a. S. Sakai, On a conjecture of Kaplansky, Tôhoku Math. J. 12 (1960), 31-33. 19b. S. Sakai, The theory of unbounded derivatives in C*-algebras, Lecture Notes, Univ. of Newcastle, England, 1977. 20. I. E. Segal, Notes towards the construction of nonlinear relativistic quantum fields. III, Properties of the C*-dynamics for a certain class of interactions, Bull. Amer. Math. Soc. 75 (1969), 1390-1395. 21. Ja. G. Sinaĭ and A. Ja. Helmskiĭ, A description of differentiations in algebras of the type of local observables of spin systems, Funktsional Anal. i Prilozhen 6 (1972), 99-100; English transl., Funct. Anal. Appl. 6 (1973), 343-344. 22. H. Takai, On a problem of Sakai in unbounded derivations, J. Funct. Anal. 43 (1981), 202-208. 23. C. -T. Yang, Hilbert’s fifth problem and related problems on transformation groups, Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, R. I., 1976, pp. 142-146.