[Paper Review] Attempts to define a Baum--Connes map via localization of categories for inverse semigroups

Bernhard Burgstaller|arXiv (Cornell University)|Jun 28, 2015

Advanced Operator Algebra Research19 references4 citations

TL;DR

This paper attempts to generalize the Meyer-Nest approach to the Baum–Connes conjecture from groups to unital, countable inverse semigroups by using triangulated category methods and localization. It establishes key tools like Bott periodicity, induction functors, and a novel $L^2(G)$-space model, but identifies a critical flaw in the duality between induction and restriction functors, leaving the existence of a Dirac element—and thus the full Baum–Connes map—conditional on unverified assumptions about a correct right adjoint to induction.

ABSTRACT

Meyer and Nest showed that the Baum--Connes map is equivalent to a map on $K$-theory of two different crossed products. This approach is strongly categorial in method since its bases is to regard Kasparov's theory $KK^G$ as a triangulated category. We have tried to translate this approach to the realm of inverse semigroup equivariant $C^*$-algebras but can prove the existence of a Baum--Connes map only under some unverified additional assumptions which we however strongly motivate. Some of our results may be of independent interest, for example Bott periodicity, the definition of induction functors, the definition of a completely novel compatible $L^2(G)$-space, a Cuntz picture of $KK^G$, and the verification that $KK^G$ is a triangulated category.

Motivation & Objective

• To extend the Meyer-Nest framework for the Baum–Connes conjecture from groups to unital, countable inverse semigroups using categorical methods.
• To define a Dirac element in $KK^G$-theory for inverse semigroups via localization, analogous to the group case.
• To construct a $C^*$-algebraic model of $\ell^2(G)$ compatible with the $KK^G$-category structure under the $E$-continuity condition.
• To verify that $KK^G$ is a triangulated category for inverse semigroups, using a Cuntz picture of morphisms.
• To define induction and restriction functors for finite subinverse semigroups and analyze their adjoint relationships.

Proposed method

• Adapts the Meyer-Nest strategy by modeling the Baum–Connes map as a $K$-theory map induced by a descent of a Dirac morphism in $KK^G$.
• Establishes $KK^G$ as a triangulated category using a Cuntz picture of $KK^G$-morphisms as $*$-homomorphisms.
• Introduces a new $L^2(G)$-space model under the $E$-continuity condition to support the construction of the Dirac element.
• Applies Brown’s representability theorem in triangulated categories to construct a candidate Dirac element in $KK^G(P, \mathbb{C})$.
• Uses a ${\mathcal{C}}{\mathcal{J}}$-simplicial approximation construction to build a candidate morphism $D$ with isomorphism properties on $KK^H$-groups.
• Replaces the classical restriction functor with a correct right adjoint to induction, based on theoretical results from Neeman, and analyzes its properties under unverified assumptions.

Experimental results

Research questions

• RQ1 Can the Meyer-Nest approach to the Baum–Connes conjecture be generalized from groups to inverse semigroups using triangulated category techniques?
• RQ2 Does the existence of a Dirac element in $KK^G$ for inverse semigroups follow from a ${\mathcal{C}}{\mathcal{J}}$-simplicial approximation under suitable assumptions?
• RQ3 What is the correct right adjoint functor to the induction functor $\mathrm{Ind}_H^G$ for finite subinverse semigroups $H \subseteq G$?
• RQ4 How can a compatible $L^2(G)$-space be constructed in the context of inverse semigroup $C^*$-algebras to support $KK^G$-theory?
• RQ5 To what extent do the classical identities like $KK^G(\mathrm{Ind}_H^G A, B) \cong KK^H(A, \mathrm{Res}_G^H B)$ hold for inverse semigroups?

Key findings

• The paper establishes that $KK^G$ for a unital, countable inverse semigroup $G$ is a triangulated category, using a Cuntz picture of morphisms.
• Bott periodicity is verified for $KK^G$-theory in the context of inverse semigroups.
• A new $L^2(G)$-space model is constructed under the $E$-continuity condition, which is essential for the $KK^G$-category structure.
• The induction functor $\mathrm{Ind}_H^G: KK^H \to KK^G$ is defined for finite subinverse semigroups $H \subseteq G$, and its right adjoint is identified as a theoretical object via Neeman’s results.
• A candidate Dirac morphism $D \in KK^G(P, \mathbb{C})$ is constructed under unverified assumptions on the correct right adjoint functor, which would imply the existence of a Baum–Connes map.
• The classical duality $KK^G(\mathrm{Ind}_H^G A, B) \cong KK^H(A, \mathrm{Res}_G^H B)$ fails in the inverse semigroup setting, and the restriction functor is not the correct right adjoint.