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The uniqueness theorem for Kasparov theory

Gábor Szabó

TL;DR

The paper proves a broad uniqueness theorem for KK-theory by establishing $K_1$-injectivity of Paschke-type dual algebras and developing a unifying framework for generalized equivariant KK-theory. It introduces an umbrella construction $KK^G(\mathfrak{C};\alpha,\beta)$ based on eligible sets of cocycle representations, shows that $\,\mathfrak{C}$-homotopy coincides with stabilization by absorbing elements, and proves a KK-uniqueness result: if two absorbing cocycle representations form an anchored $(\mathfrak{C};\alpha,\beta)$-Cuntz pair and represent zero in $KK^G(\mathfrak{C};\alpha,\beta)$, then they are strongly asymptotically unitarily equivalent. A generalized Paschke duality is developed, identifying $K_1$-groups of Paschke-type algebras with the corresponding $KK^G$-groups, thereby unifying various flavors of KK-theory (nuclear, ideal-related, equivariant) under one conceptual framework. The results have potential applications in C*-algebra classification and dynamical KK-theory, enabling robust uniqueness arguments across a wide range of settings without auxiliary stabilization in many cases.

Abstract

Answering a question of Carrión et al in their recent landmark paper on C*-algebra classification, we prove a general uniqueness theorem for $KK$-theory. Given arbitrary separable C*-algebras $A$ and $B$ and a Cuntz pair consisting of two absorbing representations $\varphi,ψ: A\to\mathcal{M}(B\otimes\mathcal{K})$, the induced element of $KK(A,B)$ vanishes if and only if $\varphi$ and $ψ$ are strongly asymptotically unitarily equivalent. This improves upon the Lin-Dadarlat-Eilers stable uniqueness theorem. The conclusion is deduced by first showing the $K_1$-injectivity of an auxiliary C*-algebra associated to the C*-pair $(A,B)$, which is sometimes called the Paschke dual algebra in the literature. Most of the article is concerned with the treatment of an umbrella theorem, which yields such a uniqueness theorem for other variants of $KK$-theory. This encompasses nuclear $KK$-theory, ideal-related $KK$-theory, equivariant $KK$-theory, or any combinations thereof.

The uniqueness theorem for Kasparov theory

TL;DR

The paper proves a broad uniqueness theorem for KK-theory by establishing -injectivity of Paschke-type dual algebras and developing a unifying framework for generalized equivariant KK-theory. It introduces an umbrella construction based on eligible sets of cocycle representations, shows that -homotopy coincides with stabilization by absorbing elements, and proves a KK-uniqueness result: if two absorbing cocycle representations form an anchored -Cuntz pair and represent zero in , then they are strongly asymptotically unitarily equivalent. A generalized Paschke duality is developed, identifying -groups of Paschke-type algebras with the corresponding -groups, thereby unifying various flavors of KK-theory (nuclear, ideal-related, equivariant) under one conceptual framework. The results have potential applications in C*-algebra classification and dynamical KK-theory, enabling robust uniqueness arguments across a wide range of settings without auxiliary stabilization in many cases.

Abstract

Answering a question of Carrión et al in their recent landmark paper on C*-algebra classification, we prove a general uniqueness theorem for -theory. Given arbitrary separable C*-algebras and and a Cuntz pair consisting of two absorbing representations , the induced element of vanishes if and only if and are strongly asymptotically unitarily equivalent. This improves upon the Lin-Dadarlat-Eilers stable uniqueness theorem. The conclusion is deduced by first showing the -injectivity of an auxiliary C*-algebra associated to the C*-pair , which is sometimes called the Paschke dual algebra in the literature. Most of the article is concerned with the treatment of an umbrella theorem, which yields such a uniqueness theorem for other variants of -theory. This encompasses nuclear -theory, ideal-related -theory, equivariant -theory, or any combinations thereof.
Paper Structure (7 sections, 23 theorems, 79 equations)

This paper contains 7 sections, 23 theorems, 79 equations.

Table of Contents

  1. Preliminaries
  2. Some cocycle representation theory
  3. The Cuntz--Thomsen picture of equivariant $KK$-theory
  4. $K_1$-injectivity of certain auxiliary $\mathrm{C}^*$-algebras
  5. On a generalized equivariant $KK$-group construction
  6. $KK$-uniqueness
  7. A generalized Paschke-type duality

Key Result

Theorem 1

Let $A$ and $B$ be separable $\mathrm{C}^*$-algebras. If $\varphi: A\to\mathcal{M}(B\otimes\mathcal{K})$ is (unitally) absorbing, then $\mathscr{D}_\varphi$ and all its quotients are $K_1$-injective.

Theorems & Definitions (62)

  • Theorem 1
  • Theorem 2
  • Lemma 1: see Kasparov88 and its proof
  • Definition 1: see Szabo21cc
  • Definition 2: cf. Thomsen98
  • Definition 3
  • Proposition 1
  • proof
  • Definition 4
  • Definition 5
  • ...and 52 more