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On the $L^p$ Baum-Connes conjecture

By Gennadi Kasparov

Also appears in collection : Conference on noncommutative geometry / Conférence de géométrie non commutative

The right side of the Baum-Connes conjecture is the $K$-theory of the reduced $C^²$-algebra $C^²_{red} (G)$ 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 $C^{²,p}_{red}(G)$. 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.

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Citation data

  • DOI 10.24350/CIRM.V.18884903
  • Cite this video Kasparov, Gennadi (05/11/2015). On the $L^p$ Baum-Connes conjecture. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.18884903
  • URL https://dx.doi.org/10.24350/CIRM.V.18884903

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