Morita equivalence and Pedersen ideals
HTML articles powered by AMS MathViewer
- by Pere Ara PDF
- Proc. Amer. Math. Soc. 129 (2001), 1041-1049 Request permission
Abstract:
We show that two $C^{*}$-algebras are strongly Morita equivalent if and only if their Pedersen ideals are Morita equivalent as rings with involution.References
- Gene D. Abrams, Morita equivalence for rings with local units, Comm. Algebra 11 (1983), no. 8, 801–837. MR 695890, DOI 10.1080/00927878308822881
- Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, 2nd ed., Graduate Texts in Mathematics, vol. 13, Springer-Verlag, New York, 1992. MR 1245487, DOI 10.1007/978-1-4612-4418-9
- P. N. Ánh and L. Márki, Morita equivalence for rings without identity, Tsukuba J. Math. 11 (1987), no. 1, 1–16. MR 899719, DOI 10.21099/tkbjm/1496160500
- P. Ara, Morita equivalence for rings with involution, Algebras and Representation Theory 2 (1999), 227–247.
- Walter Beer, On Morita equivalence of nuclear $C^{\ast }$-algebras, J. Pure Appl. Algebra 26 (1982), no. 3, 249–267. MR 678523, DOI 10.1016/0022-4049(82)90109-8
- D. Blecher, On Morita’s Fundamental Theorem for $C^{*}$-algebras, to appear in Math. Scand.
- Lawrence G. Brown, Philip Green, and Marc A. Rieffel, Stable isomorphism and strong Morita equivalence of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363. MR 463928, DOI 10.2140/pjm.1977.71.349
- Lawrence G. Brown, James A. Mingo, and Nien-Tsu Shen, Quasi-multipliers and embeddings of Hilbert $C^\ast$-bimodules, Canad. J. Math. 46 (1994), no. 6, 1150–1174. MR 1304338, DOI 10.4153/CJM-1994-065-5
- José Luis García and Juan Jacobo Simón, Morita equivalence for idempotent rings, J. Pure Appl. Algebra 76 (1991), no. 1, 39–56. MR 1140639, DOI 10.1016/0022-4049(91)90096-K
- Philip Green, The local structure of twisted covariance algebras, Acta Math. 140 (1978), no. 3-4, 191–250. MR 493349, DOI 10.1007/BF02392308
- G. G. Kasparov, Hilbert $C^{\ast }$-modules: theorems of Stinespring and Voiculescu, J. Operator Theory 4 (1980), no. 1, 133–150. MR 587371
- Shoji Kyuno, Equivalence of module categories, Math. J. Okayama Univ. 28 (1986), 147–150 (1987). MR 885023
- E. C. Lance, Hilbert $C^*$-modules, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. A toolkit for operator algebraists. MR 1325694, DOI 10.1017/CBO9780511526206
- A. J. Lazar and D. C. Taylor, Multipliers of Pedersen’s ideal, Mem. Amer. Math. Soc. 5 (1976), no. 169, iii+111. MR 412823, DOI 10.1090/memo/0169
- Nobuo Nobusawa, $\Gamma$-rings and Morita equivalence of rings, Math. J. Okayama Univ. 26 (1984), 151–156. MR 779788
- M. Parvathi and A. Ramakrishna Rao, Morita equivalence for a larger class of rings, Publ. Math. Debrecen 35 (1988), no. 1-2, 65–71. MR 971953
- Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
- N. Christopher Phillips, A new approach to the multipliers of Pedersen’s ideal, Proc. Amer. Math. Soc. 104 (1988), no. 3, 861–867. MR 929419, DOI 10.1090/S0002-9939-1988-0929419-1
- Marc A. Rieffel, Induced representations of $C^{\ast }$-algebras, Advances in Math. 13 (1974), 176–257. MR 353003, DOI 10.1016/0001-8708(74)90068-1
- Marc A. Rieffel, Morita equivalence for $C^{\ast }$-algebras and $W^{\ast }$-algebras, J. Pure Appl. Algebra 5 (1974), 51–96. MR 367670, DOI 10.1016/0022-4049(74)90003-6
- Marc A. Rieffel, Morita equivalence for operator algebras, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 285–298. MR 679708
- Bo Stenström, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer-Verlag, New York-Heidelberg, 1975. An introduction to methods of ring theory. MR 0389953, DOI 10.1007/978-3-642-66066-5
Additional Information
- Pere Ara
- Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
- MR Author ID: 206418
- Email: para@mat.uab.es
- Received by editor(s): March 30, 1998
- Received by editor(s) in revised form: June 25, 1999
- Published electronically: October 4, 2000
- Additional Notes: This work was partially supported by DGYCIT and the Comissionat per Universitats i Recerca de la Generalitat de Catalunya
- Communicated by: David R. Larson
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1041-1049
- MSC (2000): Primary 46L08, 16D90
- DOI: https://doi.org/10.1090/S0002-9939-00-05688-4
- MathSciNet review: 1814143