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Multi angle  On the $L^p$ Baum-Connes conjecture
Kasparov, Gennadi (Auteur de la Conférence) | CIRM (Editeur )

The right side of the Baum-Connes conjecture is the $K$-theory of the reduced $C^*$-algebra $C^*_{red} (G)$ of the group $G$. This algebra is the completion of the algebra $L^1(G)$ in the norm of the algebra of operators acting on $L^2(G)$. If we complete the algebra $L^1(G)$ in the norm of the algebra of operators acting on $L^p(G)$ we will get the Banach algebra $C^{*,p}_{red}(G)$. The $K$-theory of this algebra serves as the right side of the $L^p$-version of the Baum-Connes conjecture. The construction of the left side and the assembly map in this case requires a little bit of techniques of asymptotic morphisms for Banach algebras. A useful category of Banach algebras for this purpose includes all algebras of operators acting on $L^p$-spaces (which may be called $L^p$-algebras).
The current joint work in progress with Guoliang Yu aims at proving the following result:
The $L^p$-version of the Baum-Connes conjecture with coefficients in any $L^p$-algebra is true for any discrete group $G$ which admits an affine-isometric, metrically proper action on the space $X = l^p(Z)$, where $Z$ is a countable discrete set, so that the linear part of this action is induced by a measure-preserving action of $G$ on $Z$.
I will discuss the techniques involved in this work.
The right side of the Baum-Connes conjecture is the $K$-theory of the reduced $C^*$-algebra $C^*_{red} (G)$ of the group $G$. This algebra is the completion of the algebra $L^1(G)$ in the norm of the algebra of operators acting on $L^2(G)$. If we complete the algebra $L^1(G)$ in the norm of the algebra of operators acting on $L^p(G)$ we will get the Banach algebra $C^{*,p}_{red}(G)$. The $K$-theory of this algebra serves as the right side of the ...

19K35 ; 46L80 ; 58B34

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We propose a yet another definition of KR-groups, which combines those of Atiyah and Karoubi and gives a simple proof of the Bott periodicity. Using the new definition, we can formulate the bulk-edge correspondence for free fermion systems as the functoriality of the Gysin map.
This is joint work with M. Furuta, S. Hayashi, M. Kotani, Y. Kubota, and K. Sato.

19K35 ; 53C80 ; 81V70

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