Elliptic complexes over $C^{\ast }$-algebras of compact operators

Abstract

For a $C^{\ast }$-algebra $A$ of compact operators and a compact manifold $M,$ we prove that the Hodge theory holds for $A$-elliptic complexes of pseudodifferential operators acting on smooth sections of finitely generated projective $A$-Hilbert bundles over $M.$ For these $C^{\ast }$-algebras, we get also a topological isomorphism between the cohomology groups of an $A$-elliptic complex and the space of harmonic elements. Consequently, the cohomology groups appear to be Banach spaces. We prove as well, that if the Hodge theory holds for a complex in the category of Hilbert $A$-modules and continuous adjointable Hilbert $A$-module homomorphisms, the complex is self-adjoint parametrix possessing.

21 pages, 1 figure

References in corpus (1)

• Homological Algebra for Superalgebras of Differentiable Functions

Cited by in corpus (2)

• On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for $C^{\ast }$-Dynamical Systems
• Induced $C^{\ast }$-complexes in metaplectic geometry