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March 25, 2015 By

On coarse spectral geometry in even dimension

C. R. Math. Rep. Acad. Sci. Canada Vol. 33 (2) 2011, pp. 57–64

June 30, 2011

Robert Yuncken, Laboratoire de Mathematiques, Universite Blaise Pascal, Clermont-Ferrand II, CampusUniversitaire des Cezeaux, 63177 Aubiere cedex

Abstract/Résumé:

Let \(\sigma\) be the involution of the Roe algebra \(C^*|\mathbf{R}|\) which is induced from the reflection \(\mathbf{R}\colon \mathbf{R}\); \(x\mapsto -x\). A graded Fredholm module over a separable \(C^*\)-algebra \(A\) gives rise to a homomorphism \(\tilde{\rho} \colon A\colon C^*|\mathbf{R}|^\sigma\) to the fixed-point subalgebra. We use this observation to give an even-dimensional analogue of a result of Roe. Namely, we show that the \(K\)-theory of this symmetric Roe algebra is \(K_0 (C^*|\mathbf{R}|^\sigma) \cong \mathbf{Z}\), \(K_1(C^*|\mathbf{R}|^\sigma) = 0\), and that the induced map \(\tilde{\rho}_* \colon K_0(A) \colon \mathbf{Z}\) on \(K\)-theory gives the index pairing of \(K\)-homology with \(K\)-theory.

Soit \(\sigma\) l’involution de l’algèbre de Roe \(C^*|\mathbf{R}|\) induite par la réflexion \(\mathbf{R}\colon \mathbf{R}\); \(x\mapsto -x\). Un module de Fredholm gradué sur une \(C^*\)-algèbre séparable \(A\) donne lieu à un homomorphisme \(\tilde{\rho} \colon A\colon C^*|\mathbf{R}|^\sigma\) à valeurs dans la sous-algèbre des éléments invariants. En utilisant cette observation, nous montrons un analogue en dimension paire d’un résultat de Roe. Plus précisément, nous montrons que la \(K\)-théorie de cette algèbre de Roe symétrique est \(K_0 (C^*|\mathbf{R}|^\sigma) \cong \mathbf{Z}\), \(K_1(C^*|\mathbf{R}|^\sigma) = 0\) et que l’application induite \(\tilde{\rho}_* \colon K_0(A) \to \mathbf{Z}\) coïncide avec l’accouplement entre \(K\)-homologie et \(K\)-théorie.

Keywords:
AMS Subject Classification: 58G12

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