Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Duality theory for covariant systems


Author: Magnus B. Landstad
Journal: Trans. Amer. Math. Soc. 248 (1979), 223-267
MSC: Primary 46L55
DOI: https://doi.org/10.1090/S0002-9947-1979-0522262-6
MathSciNet review: 522262
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If $ (A,\rho ,G)$ is a covariant system over a locally compact group G, i.e. $ \rho $ is a homomorphism from G into the group of $ ^{\ast}$-automorphisms of an operator algebra A, there is a new operator algebra $ \mathfrak{A}$ called the covariance algebra associated with $ (A,\rho ,G)$. If A is a von Neumann algebra and $ \rho $ is $ \sigma $-weakly continuous, $ \mathfrak{A}$ is defined such that it is a von Neumann algebra. If A is a $ {C^{\ast}}$-algebra and $ \rho $ is norm-continuous $ \mathfrak{A}$ will be a $ {C^{\ast}}$-algebra. The following problems are studied in these two different settings: 1. If $ \mathfrak{A}$ is a covariance algebra, how do we recover A and $ \rho $? 2. When is an operator algebra $ \mathfrak{A}$ the covariance algebra for some covariant system over a given locally compact group G?


References [Enhancements On Off] (What's this?)

  • [1] C. A. Akemann, G. K. Pedersen and J. Tomiyama, Multipliers on $ {C^{\ast}}$-algebras, J. Functional Analysis 13 (1973), 277-301. MR 0470685 (57:10431)
  • [2] H. Araki and J. Woods, A classification of factors, Publ. Res. Inst. Math. Sci. 4 (1968), 51-130. MR 0244773 (39:6087)
  • [3] R. C. Busby, Double centralizers and extensions of $ {C^{\ast}}$-algebras, Trans. Amer. Math. Soc. 132 (1968), 79-99. MR 37 #770. MR 0225175 (37:770)
  • [4] F. Combes, Poids sur une $ {C^{\ast}}$-algèbre, J. Math. Pures Appl. (9) 47 (1968), 57-100. MR 38 #5016. MR 0236721 (38:5016)
  • [5] -, Poids associé à une algèbre hilbertienne à gauche, Compositio Math. 23 (1971), 49-77. MR 44 #5786. MR 0288590 (44:5786)
  • [6] A. Connes, Une classification de facteurs de type III, Ann. Sci. École Norm. Sup. 6 (1973), 133-252. MR 49 #5865. MR 0341115 (49:5865)
  • [7] A. Derighetti, Some results on the Fourier-Stieltjes algebra of a locally compact group, Comment. Math. Helv. 45 (1970), 219-228. MR 0412735 (54:856)
  • [8] J. Dixmier, Les algèbres d'opérateurs dans l'espace hilbertien, 2nd ed., Gauthier-Villars, Paris, 1969.
  • [9] -, Les $ {C^{\ast}}$-algèbres et leurs représentations, Gauthier-Villars, Paris, 1969. MR 30 #1404. MR 0246136 (39:7442)
  • [10] S. Doplicher, D. Kastler and D. Robinson, Covariance algebras in field theory and statistical mechanics, Comm. Math. Phys. 3 (1966), 1-28. MR 0205095 (34:4930)
  • [11] E. G. Effros and F. Hahn, Locally compact transformation groups and $ {C^{\ast}}$-algebras, Mem. Amer. Math. Soc. No. 75 (1967). MR 37 #2895. MR 0227310 (37:2895)
  • [12] E. G. Effros and E. C. Lance, Tensor products of operator algebras, Advances in Math. 25 (1977), 1-34. MR 0448092 (56:6402)
  • [12] E. G. Effros and E. C. Lance, Tensor products of operator algebras (preprint). MR 0448092 (56:6402)
  • [13] J. Ernest, The enveloping algebra of a covariant system, Comm. Math. Phys. 17 (1970), 61-74. MR 43 #1553. MR 0275800 (43:1553)
  • [14] -, A duality theorem for the automorphism group of a covariant system, Comm. Math. Phys. 17 (1970), 75-90. MR 42 #8298. MR 0273419 (42:8298)
  • [15] J. M. G Fell, An extension of Mackey's method to Banach $ ^{\ast}$-algebraic bundles, Mem. Amer. Math. Soc. No. 90 (1969). MR 0259619 (41:4255)
  • [16] G. K. Pedersen and M. Takesaki, The Radon-Nikodým theorem for von Neumann algebras, Acta Math. 130 (1973), 53-87. MR 0412827 (54:948)
  • [17] R. T. Powers, Representations of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. of Math. (2) 86 (1967), 138-171. MR 36 #1989. MR 0218905 (36:1989)
  • [18] M. Rieffel, Induced representations of $ {C^{\ast}}$-algebras, Advances in Math. 13 (1974), 176-257. MR 50 #5489. MR 0353003 (50:5489)
  • [19] W. Rudin, Fourier analysis on groups, Interscience, New York, 1967. MR 0152834 (27:2808)
  • [20] S. Sakai, $ {C^{\ast}}$-algebras and $ {W^{\ast}}$-algebras, Springer-Verlag, Berlin and New York, 1971. MR 0442701 (56:1082)
  • [21] M. E. Sweedler, Hopf algebras, Benjamin, New York, 1969. MR 40 #5705. MR 0252485 (40:5705)
  • [22] M. Takesaki, Covariant representations of $ {C^{\ast}}$-algebras and their locally compact groups, Acta Math. 119 (1967), 273-303. MR 0225179 (37:774)
  • [23] -, A characterization of group algebras as a converse of Tannaka-Stinespring-Tatsuuma duality, Amer. J. Math. 91 (1969), 529-564. MR 0244437 (39:5752)
  • [24] -, Tomita's theory of modular Hilbert algebras and its applications, Lecture Notes in Math., vol. 128, Springer-Verlag, Berlin and New York, 1970.
  • [25] -, A liminal crossed product of a uniformly hyperfinite $ {C^{\ast}}$-algebra by a compact abelian group, J. Functional Analysis 7 (1971), 140-146. MR 0275184 (43:941)
  • [26] -, Duality and von Neumann algebras, Bull. Amer. Math. Soc. 77 (1971), 553-557. MR 46 #4225. MR 0305095 (46:4225)
  • [27] -, The structure of a von Neumann algebra with a homogeneous periodic state, Acta Math. 131 (1973), 79-122. MR 0438148 (55:11067)
  • [28] -, Duality for crossed products and structure of type III von Neumann algebras, Acta Math. 131 (1967), 249-310.
  • [29] J. Tomiyama, Fubini type theorems to the tensor product of $ {C^{\ast}}$-algebras, Tôhoku Math. J. 19 (1967), 213-226. MR 0218906 (36:1990)
  • [30] G. Zeller-Meier, Produits croisés d'une $ {C^{\ast}}$-algèbre par une group d'automorphismes, J Math. Pures Appl. (9) 47 (1968), 101-239. MR 39 #3329. MR 0241994 (39:3329)
  • [31] U. Haagerup, Operator valued weights and crossed products, Symposia Math. 20 (1976), 241-251. MR 0442705 (56:1086)
  • [32] H. Takai, Dualité dans les produits croisés de $ {C^{\ast}}$-algebres, C. R. Acad. Sci. Paris 278 (1974), 1041-1043. MR 49 #5863. MR 0341113 (49:5863)
  • [33] -, On a duality for crossed products of $ {C^{\ast}}$-algebras, J. Functional Analysis 19 (1975), 25-39. MR 0365160 (51:1413)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46L55

Retrieve articles in all journals with MSC: 46L55


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0522262-6
Keywords: Operator algebra, covariant system, covariance algebra
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society