Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 041, 12 pages      arXiv:1612.03559      https://doi.org/10.3842/SIGMA.2017.041

Non-Commutative Vector Bundles for Non-Unital Algebras

Adam Rennie and Aidan Sims
School of Mathematics and Applied Statistics, University of Wollongong, Northfields Ave 2522, Australia

Received December 13, 2016, in final form June 12, 2017; Published online June 16, 2017

Abstract
We revisit the characterisation of modules over non-unital $C^*$-algebras analogous to modules of sections of vector bundles. A fullness condition on the associated multiplier module characterises a class of modules which closely mirror the commutative case. We also investigate the multiplier-module construction in the context of bi-Hilbertian bimodules, particularly those of finite numerical index and finite Watatani index.

Key words: Hilbert module; vector bundle; multiplier module; Watatani index.

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References

  1. Arambašić L., Bakić D., Frames and outer frames for Hilbert $C^*$-modules, Linear Multilinear Algebra 65 (2017), 381-431, arXiv:1507.04101.
  2. Blecher D.P., Le Merdy C., Operator algebras and their modules—an operator space approach, London Mathematical Society Monographs, Oxford Science Publications, Vol. 30, The Clarendon Press, Oxford University Press, Oxford, 2004.
  3. Dixmier J., Douady A., Champs continus d'espaces hilbertiens et de $C^{\ast}$-algèbres, Bull. Soc. Math. France 91 (1963), 227-284.
  4. Echterhoff S., Raeburn I., Multipliers of imprimitivity bimodules and Morita equivalence of crossed products, Math. Scand. 76 (1995), 289-309.
  5. Fell J.M.G., Doran R.S., Representations of $^*$-algebras, locally compact groups, and Banach $^*$-algebraic bundles, Vol. 1, Basic representation theory of groups and algebras, Pure and Applied Mathematics, Vol. 125, Academic Press, Inc., Boston, MA, 1988.
  6. Kajiwara T., Pinzari C., Watatani Y., Jones index theory for Hilbert $C^*$-bimodules and its equivalence with conjugation theory, J. Funct. Anal. 215 (2004), 1-49, arXiv:math.OA/0301259.
  7. Raeburn I., Thompson S.J., Countably generated Hilbert modules, the Kasparov stabilisation theorem, and frames with Hilbert modules, Proc. Amer. Math. Soc. 131 (2003), 1557-1564.
  8. Raeburn I., Williams D.P., Morita equivalence and continuous-trace $C^*$-algebras, Mathematical Surveys and Monographs, Vol. 60, Amer. Math. Soc., Providence, RI, 1998.
  9. Rennie A., Smoothness and locality for nonunital spectral triples, $K$-Theory 28 (2003), 127-165.
  10. Rennie A., Robertson D., Sims A., The extension class and KMS states for Cuntz-Pimsner algebras of some bi-Hilbertian bimodules, J. Topol. Anal. 9 (2017), 297-327, arXiv:1501.05363.
  11. Swan R.G., Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264-277.

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