Elliptic complexes over C∗-algebras of compact operators

Abstract

For a C * -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 * -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 finitely generated projective C * -Hilbert modules and especially, 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.Math. Subj. Class. 2010: Primary 46M18; Secondary 46L08, 46M20

Induced C*-Complexes in Metaplectic Geometry

“…Consequently, the quotient topology on the cohomology groups of complexes in these categories may be non-Hausdorff and thus, the quotients may not be Hilbert A-modules in the quotient topology. (See [36] for simple examples. )…”

“…Second proof of Theorem 18. We use the Mishchenko-Fomenko theory elaborated for complexes in [35] and [36]. It is immediate to compute that the symbols s k of d Φ k are given by s k (α ⊗ s, τ ) = (τ ∧ α) ⊗ s, τ ∈ T * m M, α ⊗ s ∈ (H k R ) m (see [34]), and a matter of multilinear algebra to see that they form an exact sequence in places k = 0, .…”

“…Consequently (compactness and ellipticity), its cohomology groups are finitely generated and projective Hilbert CH-modules by Thm. 9 in Krýsl [36]. In places Z \ {0, .…”

“…Therefore, it makes sense to find conditions when this happens. This was investigated by the author in the past years (see [35], [36] or an overview in [37]).…”

“…In the connection to sheaf cohomology, Banach and Fréchet bundles are studied in the papers of Illusie [25] and Röhrl [57]. To present a sample of the broad context (foremost connected to C * -algebras, K-theory and homological algebra) in which such bundles are considered, let us mention the papers of Maeda, Rosenberg [43], Freed, Lott [16], Larraín-Hubach [39], the author [36] and Fathizadeh, Gabriel [12]. There are also works in which holomorphic Banach bundles and their sheaf cohomology groups are treated.…”