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Shintaro Nishikawa

Publications and source records attributed to Shintaro Nishikawa.

12 recordsLinked to original sources
arXiv2025

Dehn fillings, equivariant homology, and the Baum-Connes conjecture

We establish a connection between Cohen-Lyndon triples and equivariant homology theory, with a focus on the Baum-Connes conjecture. In the first part of this work, we establish an excision sequence for the classifying spaces for proper actions in equivariant homology theories. This provides a direct link between Cohen-Lyndon triples and the left-hand side of the Baum-Connes conjecture. Independently of these, we prove that the Baum-Connes conjecture with coefficients (BCC) with finite wreath products holds for all discrete hyperbolic groups, building on the monumental work of Lafforgue. Combining this with permanence properties and the work of Dahmani-Guirardel-Osin on relatively hyperbolic groups, we identify a broad class of groups, including all lattices in simple Lie groups of real rank one that satisfy the BCC with finite wreath products. This significantly broadens the scope of our first result, as Cohen-Lyndon triples arise naturally in the context of relatively hyperbolic groups, thereby connecting both sides of the Baum-Connes conjecture.

math.KT↗
arXiv2025

Equivariant $KK$-theory of Bernoulli shifts on $C^*$-algebras with approximately inner flip

Building on Enders--Schemeitat--Tikuisis' classification, we show that a separable $C^*$-algebra $A$ with approximately inner flip in the UCT class is $K$-theoretically self-absorbing if and only if for every finite group $G$, the Bernoulli shift on $A^{\otimes G}$ is $KK^G$-equivalent to the trivial action. This in particular applies to UHF-algebras of infinite type and computes the $K$-theory of the associated crossed product. Along the way, we obtain an alternative proof of Hirshberg--Winter's result that the Bernoulli shift of $G$ on a UHF-algebra of infinite type absorbs the trivial action up to conjugacy. For more general amenable groups $G$, we develop $K$-theory formulas for Bernoulli shifts on UHF-absorbing $C^*$-algebras, and establish $KK^G$-triviality for Bernoulli shifts on strongly self-absorbing $C^*$-algebras satisfying the UCT.

math.OA↗
arXiv2022

$K$-theory of non-commutative Bernoulli Shifts

For a large class of C*-algebras $A$, we calculate the $K$-theory of reduced crossed products $A^{\otimes G}\rtimes_rG$ of Bernoulli shifts by groups satisfying the Baum--Connes conjecture. In particular, we give explicit formulas for finite-dimensional C*-algebras, UHF-algebras, rotation algebras, and several other examples. As an application, we obtain a formula for the $K$-theory of reduced C*-algebras of wreath products $H\wr G$ for large classes of groups $H$ and $G$. Our methods use a generalization of techniques developed by the second named author together with Joachim Cuntz and Xin Li, and a trivialization theorem for finite group actions on UHF algebras developed in a companion paper by the third and fourth named authors.

math.OA↗
arXiv2022

Slow exponential growth representations of Sp(n, 1) at the edge of Cowling's strip

We obtain a slow exponential growth estimate for the spherical principal series representation rho_s of Lie group Sp(n, 1) at the edge (Re(s)=1) of Cowling's strip (|Re(s)|<1) on the Sobolev space H^alpha(G/P) when alpha is the critical value Q/2=2n+1. As a corollary, we obtain a slow exponential growth estimate for the homotopy rho_s (s in [0, 1]) of the spherical principal series which is required for the first author's program for proving the Baum--Connes conjecture with coefficients for Sp(n,1).

math.RT↗
arXiv2022

Sp(n,1) admits a proper 1-cocycle for a uniformly bounded representation

We verify Shalom's conjecture for the simple real-rank-one Lie group Sp(n ,1) for any n: i.e. we show that it admits a metrically proper affine action on a Hilbert space whose linear part is a uniformly bounded representation. We provide two different proofs. Both approaches crucially use results on uniformly bounded representations by Michael Cowling. The first approach is quite abstract: it uses an automatic-properness result of Shalom and requires almost no computations. The second approach is explicit: we deduce the properness of cocycles 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).

math.GR↗
arXiv2022

Crossed product approach to equivariant localization algebras

The goal of this article is to provide a bridge between the gamma element method for the Baum--Connes conjecture (the Dirac dual-Dirac method) and the controlled algebraic approach of Roe and Yu (localization algebras). For any second countable, locally compact group G, we study the reduced crossed product algebras of the representable localization algebras for proper G-spaces. We show that the naturally defined forget-control map is equivalent to the Baum--Connes assembly map for any locally compact group G and for any coefficient G-C*-algebra B. We describe the gamma element method for the Baum--Connes conjecture from this controlled algebraic perspective. As an application, we extend the recent new proof of the Baum--Connes conjecture with coefficients for CAT(0)-cubical groups to the non-cocompact setting.

math.KT↗
arXiv2020

Proper Kasparov Cycles and the Baum-Connes Conjecture

We introduce the notion of proper Kasparov cycles for Kasparov's 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.

math.KT↗
arXiv2020

Groups with Spanier-Whitehead duality

Building on work by Kasparov, we study the notion of Spanier-Whitehead K-duality for a discrete group. It is defined as duality in the KK-category between two C*-algebras which are naturally attached to the group, namely the reduced group C*-algebra and the crossed product for the group action on the universal example for proper actions. We compare this notion to the Baum-Connes conjecture by constructing duality classes based on two methods: the standard "gamma element" technique, and the more recent approach via cycles with property gamma. As a result of our analysis, we prove Spanier-Whitehead duality for a large class of groups, including Bieberbach's space groups, groups acting on trees, and lattices in Lorentz groups.

math.KT↗
arXiv2019

On the Baum-Connes Conjecture for Groups Acting on CAT(0)-Cubical Spaces

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.

math.KT↗
arXiv2019

Direct Splitting Method for the Baum-Connes Conjecture

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.

math.OA↗
arXiv2016

On the Lifting of the Dirac Elements in the Higson-Kasparov Theorem

In this thesis, we investigate the proof of the Baum-Connes Conjecture with Coefficients for a-$T$-menable groups. We will mostly and essentially follow the argument employed by N. Higson and G. Kasparov in the paper [Nigel Higson and Gennadi Kasparov. $E$-theory and $KK$-theory for groups which act properly and isometrically on Hilbert space. Invent. Math., 144(1):23-74, 2001]. The crucial feature is as follows. One of the most important point of their proof is how to get the Dirac elements (the inverse of the Bott elements) in Equivariant $KK$-Theory. We prove that the group homomorphism used for the lifting of the Dirac elements is an isomorphism in the case of our interests. Hence, we get a clear and simple understanding of the lifting of the Dirac elements in the Higson-Kasparov Theorem. In the course of our investigation, on the other hand, we point out a problem and give a fixed precise definition for the non-commutative functional calculus which is defined in the paper In the final part, we mention that the $C^*$-algebra of (real) Hilbert space becomes a $G$-$C^*$-algebra naturally even when a group $G$ acts on the Hilbert space by an affine action whose linear part is of the form an isometry times a scalar and prove the infinite dimensional Bott-Periodicity in this case by using Fell's absorption technique.

math.OA↗