On the Picard group of a continuous trace -algebra

Author:
Iain Raeburn

Journal:
Trans. Amer. Math. Soc. **263** (1981), 183-205

MSC:
Primary 46M20; Secondary 18F25, 46L05, 58G12

DOI:
https://doi.org/10.1090/S0002-9947-1981-0590419-3

MathSciNet review:
590419

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a continuous trace -algebra with paracompact spectrum , and let be the algebra of bounded continuous functions on , so that acts on in a natural way. An bimodule is an imprimitivity bimodule if it is an imprimitivity bimodule in the sense of Rieffel and the induced actions of on the left and right of agree. We denote by the group of isomorphism classes of imprimitivity bimodules under . Our main theorem asserts that . This result is well known to algebraists if is an -homogeneous -algebra with identity, and if is separable it can be deduced from two recent descriptions of the automorphism group due to Brown, Green and Rieffel on the one hand and Phillips and Raeburn on the other. Our main motivation was to provide a direct link between these two characterisations of .

**[1]**H. Bass,*Topics in algebraic**-theory*, Tata Institute of Fundamental Research, Bombay, 1967.**[2]**-,*Algebraic**-theory*, Benjamin, New York, 1968.**[3]**L. G. Brown, P. Green and M. A. Rieffel,*Stable isomorphism and strong Morita equivalence of**-algebras*, Pacific J. Math.**71**(1977), 349-363. MR**0463928 (57:3866)****[4]**F. DeMeyer and E. Ingraham,*Separable algebras over commutative rings*, Lecture Notes in Math., vol. 181, Springer-Verlag, Berlin, 1971. MR**0280479 (43:6199)****[5]**J. Dixmier,*Champs continus d'espaces hilbertiennes et de**-algèbres*. II, J. Math. Pures Appl.**42**(1963), 1-20. MR**0150608 (27:603)****[6]**-,*Ideal center of a**-algebra*, Duke Math. J.**35**(1968), 375-382. MR**0230138 (37:5703)****[7]**-,*-algebras*, North-Holland, Amsterdam, 1977.**[8]**R. Lee,*On the**-algebras of operator fields*, Indiana Univ. Math. J.**25**(1976), 303-314. MR**0410400 (53:14150)****[9]**J. Phillips and I. Raeburn,*Automorphisms of**-algebras and second Čech cohomology*, Indiana Univ. Math. J. (to appear). MR**589649 (82b:46089)****[10]**-,*Perturbations of**-algebras*. II, Proc. London Math. Soc. (to appear).**[11]**M. A. Rieffel,*Induced representation of**-algebras*, Advances in Math.**13**(1974), 176-257.**[12]**-,*Unitary representations of group extensions; an algebraic approach to the theory of Mackey and Blattner*, in Studies in Analysis, Advances in Math. Supplementary Studies, vol. 4, Academic Press, New York, 1979, pp. 43-82. MR**546802 (81h:22004)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
46M20,
18F25,
46L05,
58G12

Retrieve articles in all journals with MSC: 46M20, 18F25, 46L05, 58G12

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1981-0590419-3

Article copyright:
© Copyright 1981
American Mathematical Society