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On Finiteness Conditions in Twisted K-Theory

Abstract

The aim of this (mostly expository) article is to show a connection between the finiteness conditions arising in twisted K-theory. There are two different conditions arising naturally in two main approaches to the problem of computing the index of the appropriate family of elliptic operators (the approach of Nistor and Troitsky and the approach of Mathai, Melrose, and Singer). These conditions are formulated absolutely differently, but in some sense they should be close to each other. In this paper, we find this connection and prove the corresponding formal statement. Thereby it is shown that these conditions map to each other. This opens a possibility to synthesize these approaches. It is also shown that the finiteness condition arising in the paper of Nistor and Troitsky is a special case of the finiteness condition that appears in the paper of Emerson and Meyer, where the theorem of Nistor and Troitsky is proved not only for the case of a bundle of Lie groups, but also for the case of a general groupoid.

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References

  1. 1.

    M. Atiyah and G. Segal, “Twisted K-theory,” Ukr. Mat. Visn., No. 1 (3), 287–330 (2004).

  2. 2.

    M. F. Atiyah and I. M. Singer, “The index of elliptic operators. I,” Ann. Math. (2), 87, 484–530 (1968).

  3. 3.

    M. F. Atiyah and I. M. Singer, “The index of elliptic operators. IV,” Ann. Math. (2), 93, 119–138 (1971).

  4. 4.

    M. R. Buneci, “Groupoid C∗-algebras,” Surv. Math. Its Appl., 1, 71–98 (2006).

    MathSciNet  MATH  Google Scholar 

  5. 5.

    J. Cuntz, R. Meyer, and J. M. Rosenberg, “Topological and bivariant K-theory,” Oberwolfach Sem., 36, 173–182 (2007).

    MathSciNet  MATH  Google Scholar 

  6. 6.

    H. Emerson and R. Meyer, “Bivariant K-theory via correspondence,” Adv. Math., 225, 2883–2919 (2010).

    MathSciNet  Article  Google Scholar 

  7. 7.

    H. Emerson and R. Meyer, “Equivariant embedding theorems and topological index maps,” Adv. Math., 225, No. 5, 2840–2882 (2010).

    MathSciNet  Article  Google Scholar 

  8. 8.

    A. Grothendieck, “Le group de Brauer: I. Alg`ebres d’Azumaya et interpr´etations diverses,” Sem. N. Bourbaki, 290, 199–219 (1964–1966).

    Google Scholar 

  9. 9.

    V. Mathai, R. B. Melrose, and I. M. Singer, “The index of projective families of elliptic operators,” Geom. Topol., 9, 341–373 (2005).

    MathSciNet  Article  Google Scholar 

  10. 10.

    V. Nistor, “An index theorem for gauge-invariant families: The case of solvable groups,” Acta Math. Hungar., 99, No. 1-2, 155–183 (2003).

    MathSciNet  Article  Google Scholar 

  11. 11.

    V. Nistor and E. Troitsky, “An index for gauge-invariant operators and the Dixmier–Douady invariant,” Trans. Amer. Math. Soc., 356, No. 1, 185–218 (2004).

    MathSciNet  Article  Google Scholar 

  12. 12.

    V. Nistor and E. Troitsky, “The Thom isomorphism in gauge-equivariant K-theory,” in: C∗-Algebras and Elliptic Theory, Trends Math., Birkh¨auser (2006), pp. 213–245

  13. 13.

    V. Nistor and E. Troitsky, An Index Theorem in the Gauge-Equivariant K-Theory, http://wwwmath. uni-muenster.de/u/echters/Focused-Semester/workshopabstracts.html, http://mech.math. msu.su/~troitsky/muens.pdf (2009).

  14. 14.

    V. Nistor and E. Troitsky, “Analysis of gauge-equivariant complexes and a topological index theorem for gauge-invariant families,” Russ. J. Math. Phys., 22, Issue 1, 74–97 (2015).

    MathSciNet  Article  Google Scholar 

  15. 15.

    J. Renault, “The ideal structure of groupoid cross product C∗-algebras,” J. Operator Theor., 25, 3–36 (1991).

    MATH  Google Scholar 

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Correspondence to M. A. Gerasimova.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 5, pp. 61–77, 2016.

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Gerasimova, M.A. On Finiteness Conditions in Twisted K-Theory. J Math Sci 248, 553–563 (2020). https://doi.org/10.1007/s10958-020-04896-w

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