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abstract: ‘We show that the standard formulation of the Baum-Connes conjecture for a group with a countable abelian subgroup, $G$, implies the $r$-malleable Banach-Tarski paradox and the non-isomorphism of $G$ with the additive group of reals, $\mathbb{R}$.’
author:
– |
James Funke\
`jfunke@gmail.com`
title: ‘The Baum-Connes conjecture implies the Banach-Tarski paradox’
—
Introduction
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The Baum-Connes conjecture states that the assembly map $$\mu\colon\ \lim_{n\to\infty} KK_\ast(C(X),\mathbb{C})\longrightarrow K_\ast(C(X)\rtimes G)$$ is an isomorphism for a given group $G$. Moreover, the assembly map becomes an isomorphism after tensoring over the coefficients.
If we consider the Baum-Connes conjecture with coefficients in a simple unital C\*-algebra $A$, ie. $KK_\ast(C(X),A)$, we can see this conjecture as an extension of the Kasparov-Novikov conjecture, asserting that $KK_\ast(C(X)\rtimes G,A)$ agrees with $KK_\ast(C(X),A)$. Therefore, the Baum-Connes conjecture implies the Kasparov-Novikov conjecture. While extensions of the Kasparov-Novikov conjecture have been shown by Kasparov [@katz] and Yaskin [@yaskin], the Baum-Connes conjecture has not.
In this paper we show that the Baum-Connes conjecture implies the Banach-Tarski paradox. The Banach-Tarski paradox concerns the non-existence of a decomposition of the unit ball, $\mathbb{B}(H)$ of a Hilbert space $H$, into a sum of unit balls, ${\mathbb{B}}^n$. Aspherical groups possess an isometric circle action and the
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