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Register a new userProper Kasparov Cycles and the Baum-Connes Conjecture
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Authors
Shintaro Nishikawa
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We introduce the notion of proper Kasparov cycles for Kasparovs G-equivariant KK-theory for a general locally compact, second countable topological group G. We show that for any proper Kasparov cycle, its induced map on K-theory factors through the left-hand side of the Baum-Connes conjecture. This allows us to upgrade the direct splitting method, a recent new approach to the Baum-Connes conjecture which, in contrast to the standard gamma element method (the Dirac dual-Dirac method), avoids the need of constructing proper algebras and the Dirac and the dual-Dirac elements. We introduce the notion of Kasparov cycles with Property (gamma) removing the G-compact assumption on the universal space EG in the previous paper Direct Splitting Method for the Baum-Connes Conjecture. We show that the existence of a cycle with Property (gamma) implies the split-injectivity of the Baum-Connes assembly map for all coefficients. We also obtain results concerning the surjectivity of the assembly map.
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Let $left( 1to N_nto G_nto Q_nto 1 right)_{nin mathbb{N}}$ be a sequence of extensions of finitely generated groups with uniformly finite generating subsets. We show that if the sequence $left( N_n right)_{nin mathbb{N}} $ with the induced metric from the word metrics of $left( G_n right)_{nin mathbb{N}} $ has property A, and the sequence $left( Q_n right)_{nin mathbb{N}} $ with the quotient metrics coarsely embeds into Hilbert space, then the coarse Baum-Connes conjecture holds for the sequence $left( G_n right)_{nin mathbb{N}}$, which may not admit a coarse embedding into Hilbert space. It follows that the coarse Baum-Connes conjecture holds for the relative expanders and group extensions exhibited by G. Arzhantseva and R. Tessera, and special box spaces of free groups discovered by T. Delabie and A. Khukhro, which do not coarsely embed into Hilbert space, yet do not contain a weakly embedded expander. This in particular solves an open problem raised by G. Arzhantseva and R. Tessera cite{Arzhantseva-Tessera 2015}.
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 introduce a new method for studying the Baum-Connes conjecture, which we call the direct splitting method. The method can simplify and clarify proofs of some of the known cases of the conjecture. In a separate paper, with J. Brodzki, E. Guentner and N. Higson, a similar idea will be used 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.
We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group $Gamma$ implies that $Gamma$ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition.
We define and study an analogue of the Baum-Connes assembly map for complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups. Our starting point is the deformation picture of the Baum-Connes assembly map for a complex semisimple Lie group $ G $, which allows one to express the $ K $-theory of the reduced group $ C^* $-algebra of $ G $ in terms of the $ K $-theory of its associated Cartan motion group. The latter can be identified with the semidirect product of the maximal compact subgroup $ K $ acting on $ mathfrak{k}^* $ via the coadjoint action. In the quantum case the role of the Cartan motion group is played by the Drinfeld double of the classical group $ K $, whose associated group $ C^* $-algebra is the crossed product of $ C(K) $ with respect to the adjoint action of $ K $. Our quantum assembly map is obtained by varying the deformation parameter in the Drinfeld double construction applied to the standard deformation $ K_q $ of $ K $. We prove that the quantum assembly map is an isomorphism, thus providing a description of the $ K $-theory of complex quantum groups in terms of classical topology. Moreover, we show that there is a continuous field of $ C^* $-algebras which encodes both the quantum and classical assembly maps as well as a natural deformation between them. It follows in particular that the quantum assembly map contains the classical Baum-Connes assembly map as a direct summand.
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