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Topology of KK-theory via inverse limits of discrete abelian groups

Arturo Jaime

TL;DR

The work addresses the topology of KK-theory groups for separable $C^{*}$-algebras by leveraging the Milnor exact sequence from Willett–Yu’s controlled KK-theory. It establishes that the Rørdam group $KL(A,B)$ is pro-countable and totally disconnected, and that $KK(A,B)$ is topologically the product of $KL(A,B)$ with the closure of zero, with the full topology governed by Mittag-Leffler conditions on corresponding inverse systems. A thorough inverse-limit analysis shows these limits are Polish, and a detailed classification yields ten possible topologies for $KK(A,B)$ depending on stabilization phenomena. The results connect abstract inverse-limit theory with concrete KK-theory topology, and answer questions about the disconnectedness of the Rørdam group in this framework.

Abstract

This paper seeks to characterize some topological properties of pro-countable abelian topological groups. Using the Milnor exact sequence given by the controlled picture of $KK$-theory by Willett and Yu, we describe topological properties of the topological group $KK(A,B)$ with respect to the satisfaction of Mittag-Leffler and stability conditions of certain inverse systems.

Topology of KK-theory via inverse limits of discrete abelian groups

TL;DR

The work addresses the topology of KK-theory groups for separable -algebras by leveraging the Milnor exact sequence from Willett–Yu’s controlled KK-theory. It establishes that the Rørdam group is pro-countable and totally disconnected, and that is topologically the product of with the closure of zero, with the full topology governed by Mittag-Leffler conditions on corresponding inverse systems. A thorough inverse-limit analysis shows these limits are Polish, and a detailed classification yields ten possible topologies for depending on stabilization phenomena. The results connect abstract inverse-limit theory with concrete KK-theory topology, and answer questions about the disconnectedness of the Rørdam group in this framework.

Abstract

This paper seeks to characterize some topological properties of pro-countable abelian topological groups. Using the Milnor exact sequence given by the controlled picture of -theory by Willett and Yu, we describe topological properties of the topological group with respect to the satisfaction of Mittag-Leffler and stability conditions of certain inverse systems.
Paper Structure (6 sections, 22 theorems, 50 equations, 2 tables)

This paper contains 6 sections, 22 theorems, 50 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Inverse Limits of Topological Groups
  3. Metrizability
  4. The long exact lim - lim$^{1}$ sequence
  5. Topology
  6. Application to $KK$-Theory

Key Result

Lemma 2.1

Let $G$ be an abelian topological group and $\mathcal{B}$ a basis neighborhood for $0 \in G$. Then for any $g \in G$, the set is a basis neighborhood around $g$.

Theorems & Definitions (48)

  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Remark 2.1
  • Proposition 2.5
  • proof
  • Remark 2.2
  • Definition 3.1
  • ...and 38 more