We develop a new method for studying the Baum-Connes conjecture, which we call the direct splitting method. We introduce what we call property (γ) for G-equivariant Kasparov cycles. We show that the existence of a G-equivariant Kasparov cycle with property (γ) implies the split-injectivity of the assembly map µ G A for any separable G-C * -algebra A. We also show that if such a cycle exists, the assembly map µ G A is an isomorphism if and only if the cycle acts as the identity on the right-hand side group K * (A ⋊ r G) of the Baum-Connes conjecture. In a separate paper, with J. Brodzki, E. Guentner and N. Higson, we use this method to give a finite-dimensional proof of the Baum-Connes conjecture for groups which act properly and co-compactly on a finite-dimensional CAT(0)-cubical space.

A three-component hydrocarboxylation of an olefin with CO2 and H2 could be regarded as a dream reaction, since it would provide a straightforward approach for the synthesis of aliphatic carboxylic acids in perfect atom economy. However, this transformation has not been realized in a direct manner under mild conditions, because boosting the carboxylation with thermodynamically stable CO2 while suppressing the rapid hydrogenation of olefin remains a challenging task. Here, we report a rhodium-catalysed reductive hydrocarboxylation of styrene derivatives with CO2 and H2 under mild conditions, in which H2 served as the terminal reductant. In this approach, the carboxylation process was largely accelerated by visible light irradiation, which was proved both experimentally and by computational studies. Hydrocarboxylation of various kinds of styrene derivatives was achieved in good yields without additional base under ambient pressure of CO2/H2 at room temperature. Mechanistic investigations revealed that use of a cationic rhodium complex was critical to achieve high hydrocarboxylation selectivity.

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We give a new proof of the Baum-Connes conjecture with coefficients for any second countable, locally compact topological group that acts properly and cocompactly on a finite-dimensional CAT(0)-cubical space with bounded geometry. The proof uses the Julg-Valette complex of a CAT(0)-cubical space introduced by the first three authors, and the direct splitting method in Kasparov theory developed by the last author.

We show that the simple rank one Lie group Sp(n, 1) for any n admits a proper 1-cocycle for a uniformly bounded Hilbert space representation: i.e. it admits a metrically proper affine action on a Hilbert space whose linear part is a uniformly bounded representation. Our construction is a simple modification of the one given by Pierre Julg but crucially uses results on uniformly bounded representations by Michael Cowling. An interesting new feature is that the properness of these cocycles follows from the non-continuity of a critical case of the Sobolev embedding. This work is inspired from Pierre Julg's work on the Baum-Connes conjecture for Sp(n, 1).