no. 3
A C * -algebraic Schoenberg theorem
Annales de l'Institut Fourier, Tome 34 (1984) no. 3, pp. 155-187.

Soient 𝔄 une C * -algèbre, G un groupe compact abélien, τ une action de G sur 𝔄,𝔄 τ la sous-algèbre des points fixes de τ et 𝔄 F la sous-algèbre dense de 𝔄, des éléments G-finis. Soit ensuite H un opérateur linéaire de 𝔄 F dans 𝔄 qui commute avec τ et qui est nul sur 𝔄 τ . Nous prouvons que H est une dissipation complète si et seulement si H est fermable et sa clôture est le générateur d’un semi-groupe de type C 0 de contractions complètement positives. Ces dissipations complètes sont classifiées à l’aide de certaines applications de type négatif tordu du groupe dual G ^ dans des opérateurs dissipatifs, affiliés au centre de l’algèbre des multiplicateurs de 𝔄 τ . Dans ce cadre, il est également établi que les dissipations complètes forment un sous-ensemble propre des dissipations générales, sauf pour le cas où 𝔄 est une C * -algèbre abélienne.

Let 𝔄 be a C * -algebra, G a compact abelian group, τ an action of G by *-automorphisms of 𝔄,𝔄 τ the fixed point algebra of τ and 𝔄 F the dense sub-algebra of G-finite elements in 𝔄. Further let H be a linear operator from 𝔄 F into 𝔄 which commutes with τ and vanishes on 𝔄 τ . We prove that H is a complete dissipation if and only if H is closable and its closure generates a C 0 -semigroup of completely positive contractions. These complete dissipations are classified in terms of certain twisted negative definite maps from the dual group G ^ into dissipative operators affiliated with the center of the multiplier algebra of 𝔄 τ . We also argue that the complete dissipation property is strictly stronger than the usual dissipation property, except in special circumstances such as when 𝔄 is abelian.

@article{AIF_1984__34_3_155_0,
     author = {Bratteli, Ola and Jorgensen, Palle E. T. and Kishimoto, Akitaka and Robinson, Donald W.},
     title = {A $C^*$-algebraic {Schoenberg} theorem},
     journal = {Annales de l'Institut Fourier},
     pages = {155--187},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {34},
     number = {3},
     year = {1984},
     doi = {10.5802/aif.981},
     zbl = {0536.46046},
     mrnumber = {86b:46105},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.981/}
}
TY  - JOUR
AU  - Bratteli, Ola
AU  - Jorgensen, Palle E. T.
AU  - Kishimoto, Akitaka
AU  - Robinson, Donald W.
TI  - A $C^*$-algebraic Schoenberg theorem
JO  - Annales de l'Institut Fourier
PY  - 1984
SP  - 155
EP  - 187
VL  - 34
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.981/
DO  - 10.5802/aif.981
LA  - en
ID  - AIF_1984__34_3_155_0
ER  - 
%0 Journal Article
%A Bratteli, Ola
%A Jorgensen, Palle E. T.
%A Kishimoto, Akitaka
%A Robinson, Donald W.
%T A $C^*$-algebraic Schoenberg theorem
%J Annales de l'Institut Fourier
%D 1984
%P 155-187
%V 34
%N 3
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.981/
%R 10.5802/aif.981
%G en
%F AIF_1984__34_3_155_0
Bratteli, Ola; Jorgensen, Palle E. T.; Kishimoto, Akitaka; Robinson, Donald W. A $C^*$-algebraic Schoenberg theorem. Annales de l'Institut Fourier, Tome 34 (1984) no. 3, pp. 155-187. doi : 10.5802/aif.981. https://aif.centre-mersenne.org/articles/10.5802/aif.981/

[1] C.A. Akemann and M.E. Walter, Unbounded negative definite functions, Can J. Math., 33 (1981), 862-871. | MR | Zbl

[2] H. Araki, Normal positive linear mappings of norm 1 from a von Neumann algebra into its commutant and its application, Pub. RIMS, Kyoto, 8 (1972/1973), 439-469. | MR | Zbl

[3] W.B. Arveson, Subalgebras of C*-algebras, Acta Math., 123 (1969), 141-224. | MR | Zbl

[4] C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups, Springer-Verlag, Berlin-Heidelberg-New York, 1975. | MR | Zbl

[5] O. Bratteli, G.A. Elliot and P.E.T. Jørgensen, Decomposition of unbounded derivations into invariant and approximately inner parts, Crelle's Journal, 346 (1984), 166-193. | MR | Zbl

[6] O. Bratteli and D.E. Evans, Dynamical semigroups commuting with compact abelian actions, Ergod. Th. & Dynam. Sys., 3 (1983), 187-217. | MR | Zbl

[7] O. Bratteli and P.E.T. Jørgensen, Unbounded *-derivations and infinitesimal generators on operator algebra in Proceedings Symp. in Pure Math., Vol. 38 Part 2, 353-365, AMS Providence, R.I. (1982). | MR | Zbl

[8] O. Bratteli and P.E.T. Jørgensen, Unbounded derivations tangential to compact groups of automorphisms, J. Funct. Anal., 48 (1982), 107-133. | MR | Zbl

[9] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol I, Springer-Verlag, New York, 1979. | MR | Zbl

[10] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol. II Springer-Verlag, New York, 1981. | MR | Zbl

[11] O. Bratteli and D.W. Robinson, positive C0-semigroups on C*-algebras, Math. Scand., 49 (1981), 259-274. | MR | Zbl

[12] M.-D Choi, Some assorted inequalities for positive linear maps on C*-algebras, J. Operator Theory, 4 (1980), 271-285. | MR | Zbl

[13] E. Christensen and D.E. Evans, Cohomology of operator algebras and quantum dynamical semigroups, J. London Math. Soc. (2), 20 (1978), 358-368. | Zbl

[14] D.E. Evans, Positive linear maps on operator algebras, Commun. Math. Phys., 48 (1976), 15-22. | MR | Zbl

[15] D.E. Evans and H. Hanche-Olsen, The generators of positive semigroups, J. Funct. Anal., 32 (1979), 207-212. | MR | Zbl

[16] F.R. Gantmacher, The theory of matrices, Vol. 1, Chelsea Publishing Co., New York, 1959. | Zbl

[17] A. Kishimoto, Dissipations and derivations, Commun. Math. Phys., 47 (1976), 25-32. | MR | Zbl

[18] G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys., 48 (1976), 119-130. | MR | Zbl

[19] M.A. Naimark, Normed Algebras, Walters-Noordhoff, Groningen, 1972.

[20] D. Olsen, G.K. Pedersen and M. Takesaki, Ergodic actions of compact abelian groups, J. Operator Theory, 3 (1980), 237-269. | MR | Zbl

[21] K. Parthasarathy and K. Schmidt, Positive definite kernels, continuous tensor products, and central limit theorems of probability theory, SLN 272, Springer-Verlag, Berlin, 1972. | MR | Zbl

[22] G.K. Pedersen, C*-algebras and their Automorphism Groups, Academic Press, London, 1979. | MR | Zbl

[23] D.W. Robinson, Strongly positive semigroups and faithful invariant states, Commun. Math. Phys., 85 (1982), 129-142. | MR | Zbl

[24] S. Sakai, Developments in the theory of unbounded derivations in C*-algebras, in Operator Algebras and Applications, Proceedings of Symp. Pure Math, Vol. 38 Part 1, 309-311. AMS, Providence R.I. (1980). | MR | Zbl

[25] I.J. Schoenberg, Metric spaces and positive definite functions, Trans. Amer Math. Soc., 44 (1938), 522-536. | JFM | MR | Zbl

[26] I. Schur, Bemerkungen zur Theorie der beschränkten bilinearformen mit unendlich vielen Veränderlichen, J. für die reine u.ang. Math., 140 (1911), 1-28. | JFM

[27] I.E. Segal, A non-commutative version of abstract integration, Ann. Math., 57 (1953), 401-457. | MR | Zbl

[28] J. Slawny, On factor representations and the C*algebra of the canonical commutation relations, Commun. Math. Phys., 24 (1971), 151-170. | MR | Zbl

[29] W.F. Stinespring, Positive function on C*algebras, Proc. Amer. Math. Soc., 6 (1955), 211-216. | MR | Zbl

[30] M.H. Stone, On unbounded operators in Hilbert space, J. Indian Math. Soc., 15 (1951), 155-192. | MR | Zbl

[31] M.D. Choi, Positive linear maps on C*-algebras, Can. J. Math., 24 (1972), 520-529. | MR | Zbl

[32] A. Weil, L'intégration dans les groupes topologiques et ses applications, Hermann, Paris, 1940. | JFM | MR | Zbl

Cité par Sources :