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Elliptic operators and \(K\)-homology. (English) Zbl 1505.19004

Summary: If a differential operator \(D\) on a smooth Hermitian vector bundle \(S\) over a compact manifold \(M\) is symmetric, it is essentially self-adjoint and so admits the use of functional calculus. If \(D\) is also elliptic, then the Hilbert space of square integrable sections of \(S\) with the canonical left \(C (M)\)-action and the operator \(\chi (D)\) for \(\chi\) a normalizing function is a Fredholm module, and its \(K\)-homology class is independent of \(\chi\). In this expository article, we provide a detailed proof of this fact following the outline in the book “Analytic \(K\)-homology” by N. Higson and J. Roe [Analytic K-homology. Oxford: Oxford University Press (2000; Zbl 0968.46058)].

MSC:

19K33 Ext and \(K\)-homology
46L80 \(K\)-theory and operator algebras (including cyclic theory)
58J20 Index theory and related fixed-point theorems on manifolds
46F12 Integral transforms in distribution spaces

Citations:

Zbl 0968.46058
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References:

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