Weighted shifts and covariance algebras

O’Donovan, Donal P.

doi:10.1090/S0002-9947-1975-0385632-1

Weighted shifts and covariance algebras
HTML articles powered by AMS MathViewer

by Donal P. O’Donovan PDF

Trans. Amer. Math. Soc. 208 (1975), 1-25 Request permission

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.

References

William B. Arveson, Analyticity in operator algebras, Amer. J. Math. 89 (1967), 578–642. MR 223899, DOI 10.2307/2373237

Representations of ${C^ \ast }$-algebras

William B. Arveson and Keith B. Josephson, Operator algebras and measure preserving automorphisms. II, J. Functional Analysis 4 (1969), 100–134. MR 0250081, DOI 10.1016/0022-1236(69)90025-1
John W. Bunce and James A. Deddens, $C^{\ast }$-algebras generated by weighted shifts, Indiana Univ. Math. J. 23 (1973/74), 257–271. MR 341108, DOI 10.1512/iumj.1973.23.23022
L. A. Coburn, The $C^{\ast }$-algebra generated by an isometry, Bull. Amer. Math. Soc. 73 (1967), 722–726. MR 213906, DOI 10.1090/S0002-9904-1967-11845-7
L. A. Coburn and A. Lebow, Algebaic theory of Fredholm operators, J. Math. Mech. 15 (1966), 577–584. MR 0192343
H. O. Cordes, On a class of $C^{\ast }$-algebras, Math. Ann. 170 (1967), 283–313. MR 209853, DOI 10.1007/BF01350606
Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
Sergio Doplicher, Daniel Kastler, and Derek W. Robinson, Covariance algebras in field theory and statistical mechanics, Comm. Math. Phys. 3 (1966), 1–28. MR 205095, DOI 10.1007/BF01645459
Edward G. Effros and Frank Hahn, Locally compact transformation groups and $C^{\ast }$- algebras, Memoirs of the American Mathematical Society, No. 75, American Mathematical Society, Providence, R.I., 1967. MR 0227310
Edward G. Effros, Transformation groups and $C^{\ast }$-algebras, Ann. of Math. (2) 81 (1965), 38–55. MR 174987, DOI 10.2307/1970381
James Glimm, Locally compact transformation groups, Trans. Amer. Math. Soc. 101 (1961), 124–138. MR 136681, DOI 10.1090/S0002-9947-1961-0136681-X
A. Guichardet, Utilisation des sous-groupes distingués ouverts dans l’étude des représentations unitaires des groupes localement compacts, Compositio Math. 17 (1965), 1–35 (1965) (French). MR 188348
Paul R. Halmos, Lectures on ergodic theory, Publications of the Mathematical Society of Japan, vol. 3, Mathematical Society of Japan, Tokyo, 1956. MR 0097489
Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
Paul R. Halmos and John von Neumann, Operator methods in classical mechanics. II, Ann. of Math. (2) 43 (1942), 332–350. MR 6617, DOI 10.2307/1968872
Kenneth Hoffman and Ray Kunze, Linear algebra, Prentice-Hall Mathematics Series, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961. MR 0125849

Weighted shifts on Hilbert space

Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0054173
George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 98328, DOI 10.1007/BF02392428

Centered operators

Weighted translation operators

Masamichi Takesaki, Covariant representations of $C^{\ast }$-algebras and their locally compact automorphism groups, Acta Math. 119 (1967), 273–303. MR 225179, DOI 10.1007/BF02392085
G. Zeller-Meier, Produits croisés d’une $C^{\ast }$-algèbre par un groupe d’automorphismes, J. Math. Pures Appl. (9) 47 (1968), 101–239 (French). MR 241994