Abstract
For a class of co-chain complexes in the category of pre-Hilbert \(A\)-modules, we prove that their cohomology groups equipped with the canonical quotient topology are pre-Hilbert \(A\)-modules, and derive the Hodge theory and, in particular, the Hodge decomposition for them. As an application, we show that \(A\)-elliptic complexes of pseudodifferential operators acting on sections of finitely generated projective \(A\)-Hilbert bundles over compact manifolds belong to this class if the images of the continuous extensions of their associated Laplace operators are closed. Moreover, we prove that the cohomology groups of these complexes share the structure of the fibers, in the sense that they are also finitely generated projective Hilbert \(A\)-modules.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Denicola, D., Marcolli, M.: Zainy al-Yasry, A.: Spin foams and noncommutative geometry. Class. Quantum Grav. 27(205025) (2010)
Dixmier, J.: \(C^*\)-algebras. North-Holland Mathematical Library, vol. 15, second edn. North-Holland Publishing Co. (1982)
Drechsler, W., Tuckey, P.: On quantum and parallel transport in a Hilbert bundle over spacetime. Class. Quantum Grav. 13, 611–632 (1996)
Erat, M.: The cohomology of Banach space bundles over \(1\)-convex manifolds is not always Hausdorff. Math. Nachr. 248–249, 97–101 (2003)
Exel, R.: A Fredholm operator approach to Morita equivalence. K Theory 7, 285–308 (1993)
Fomenko, A., Mishchenko, A.: The index of elliptic operators over \(C^*\)-algebras. Izv. Akad. Nauk SSSR Ser. Mat. 43(4), 831–859 (1979)
Frank, M., Larson, D.: Frames in Hilbert \(C^*\)-modules and \(C^*\)-algebras. J. Oper. Theory 48(2), 273–314 (2002)
Krýsl, S.: Hodge theory for elliptic complexes over unital \(C^*\)-algebras. Ann. Glob. Anal. Geom. 45(3), 197–210 (2014)
Lance, C.: Hilbert \(C^*\)-modules. A toolkit for operator algebraists. London Mathematical Society Lecture Note Series, 210, Cambridge University Press, Cambridge (1995)
Palais, R.: Seminar on the Atiyah-Singer Index Theorem. Annals of Mathematical Study, No. 4, Princeton University Press (1964)
Prugovečki, E.: On quantum-geometric connections and propagators in curved spacetime. Class. Quantum Gravity 13(5), 1007–1021 (1996)
Schick, T.: \(L^2\)-index theorems, \(KK\)-theory, and connections. New York J. Math. 11, 387–443 (2005)
Solovyov, Y., Troitsky, E.: \(C^*\)-algebras and elliptic operators in differential topology. Translation of Mathematical Monographs, 192. AMS, Providence, Rhode-Island (2001)
Troitsky, E.: The index of equivariant elliptic operators over \(C^*\)-algebras. Ann. Glob. Anal. Geom. 5(1), 3–22 (1987)
Wells, R.: Differential analysis on complex manifolds. Graduate Texts in Mathematics. Springer, New York (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Krýsl, S. Hodge theory for complexes over \(C^*\)-algebras with an application to \(A\)-ellipticity. Ann Glob Anal Geom 47, 359–372 (2015). https://doi.org/10.1007/s10455-015-9449-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-015-9449-1
Keywords
- Hodge theory
- Hilbert \(C^*\)-modules
- \(C^*\)-Hilbert bundles
- Elliptic systems of partial differential equations