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Weighted shifts and covariance algebras


Author: Donal P. O’Donovan
Journal: Trans. Amer. Math. Soc. 208 (1975), 1-25
MSC: Primary 47C10; Secondary 47B37
DOI: https://doi.org/10.1090/S0002-9947-1975-0385632-1
MathSciNet review: 0385632
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Abstract: The $ {C^ \ast }$-algebras generated by bilateral and unilateral shifts are studied in terms of certain covariance algebras. This enables one to obtain an answer to the question of when such shifts are G.C.R., or not, or even when they are N.G.C.R.. In addition these shifts are classified to within algebraic equivalence.


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  • [1] W.B.Arveson, Analyticity in operator algebras, Amer. J. Math. 89 (1967), 578-642. MR 36 #6946. MR 0223899 (36:6946)
  • [2] -, Representations of $ {C^ \ast }$-algebras, Lecture Notes in Math., Springer-Verlag, New York (to appear).
  • [3] W.B. Arveson and K.B. Josephson, Operator algebras and measure preserving automorphisms. II, J. Functional Analysis 4 (1969), 100-134. MR 40 #3322. MR 0250081 (40:3322)
  • [4] J.W. Bunce and J.A. Deddens, $ {C^ \ast }$-algebras generated by weighted shifts, Indiana Univ. Math. J. 23 (1973), 257-271. MR 0341108 (49:5858)
  • [5] L.A. Coburn, The $ {C^ \ast }$-algebra generated by an isometry, Bull. Amer. Math. Soc. 73 (1967), 722-726. MR 35 #4760. MR 0213906 (35:4760)
  • [6] L.A. Coburn and A. Lebow, Algebraic theory of Fredholm operators, J. Math. Mech. 15 (1966), 577-584. MR 33 #569. MR 0192343 (33:569)
  • [7] H.O. Cordes, On a class of $ {C^ \ast }$-algebras, Math. Ann. 170 (1967), 283-313. MR 35 #749. MR 0209853 (35:749)
  • [8] J. Dixmier, Les $ {C^ \ast }$-algèbres et leurs représentations, Cahiers scientifique, fasc. 29, Gauthier-Villars, Paris, 1964. MR 30 #1404. MR 0171173 (30:1404)
  • [9] S. Doplicher, D. Kastler and D.W. Robinson, Covariance algebras in field theory and statistical mechanics, Comm. Math. Phys. 3 (1966), 1-28, MR 34 #4930. MR 0205095 (34:4930)
  • [10] 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)
  • [11] E.G. Effros, Transformation groups and $ {C^ \ast }$-algebras, Ann. of Math. (2) 81 (1965), 38-55. MR 30 #5175. MR 0174987 (30:5175)
  • [12] J.G. Glimm, Locally compact transformation groups, Trans. Amer. Math. Soc. 101 (1961), 124-138. MR 25 #146. MR 0136681 (25:146)
  • [13] A. Guichardet, Utilisation des sous-groupes distingués ouverts dans l'étude des représentations unitaires des groupes localement compacts, Compositio Math. 17 (1965), 1-35. MR 32 #5787. MR 0188348 (32:5787)
  • [14] P.R. Halmos, Lectures on ergodic theory, Publ. Math. Soc. Japan, no. 3, Math. Soc. Japan, Tokyo, 1956; reprint, Chelsea, New York, 1960. MR 20 #3958; 22 #2677. MR 0097489 (20:3958)
  • [15] -, A Hilbert space problem book, Van Nostrand, Princeton, N.J., 1967. MR 34 #8178. MR 0208368 (34:8178)
  • [16] P.R. Halmos and J. von Neumann, Operator methods in classical mechanics. II, Ann. of Math. (2) 43 (1942), 332-350. MR 4, 14. MR 0006617 (4:14e)
  • [17] K. Hoffman and R. Kunze, Linear algebra, Prentice-Hall Math. Ser., Prentice-Hall, Englewood Cliffs, N.J., 1961. MR 23 #A3146. MR 0125849 (23:A3146)
  • [18] R.L. Kelley, Weighted shifts on Hilbert space, Ph. D. Dissertation, University of Michigan, Ann Arbor, Mich., 1966.
  • [19] L.H. Loomis, An introduction to abstract harmonic analysis, Van Nostrand, Princeton, N.J., 1953. MR 14, 883. MR 0054173 (14:883c)
  • [20] G.W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265-311. MR 20 #4789. MR 0098328 (20:4789)
  • [21] B.S. Morrel and P.S. Muhly, Centered operators (preprint).
  • [22] S. K. Parrot, Weighted translation operators, Ph.D. Dissertation, University of Michigan, Ann Arbor, Mich., 1965.
  • [23] M. Takesaki, Covariant representations of $ {C^ \ast }$-algebras and their locally compact transformation groups, Acta Math. 119 (1967), 273-303. MR 0225179 (37:774)
  • [24] G. Zeller-Meier, Porduits croisés d'une $ {C^ \ast }$-algèbre par un groupe d'automorphismes, J. Math. Pures Appl. (9) 47 (1968), 101-239. MR 39 #3329. MR 0241994 (39:3329)

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DOI: https://doi.org/10.1090/S0002-9947-1975-0385632-1
Article copyright: © Copyright 1975 American Mathematical Society

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