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The universal property of graded $KK^G$-theory

Bernhard Burgstaller

Abstract

A universal category-theoretical characterization of groupoid equivariant $KK^G$-theory for ${\mathbb{Z}}_2$-graded $C^*$-algebras is established, by observing the ``$KK$-axiom'' that for each $[s,{\cal E} \oplus B, \mathbb{F}] \in KK^G(A,B)$, the `corner-embedding' $*$-homomorphism ${\bf j}: B \rightarrow {\sf cl} \big({\cal K}_B({\cal E} \oplus B) + s(A) + \mathbb{F} \cdot s(A) \big)$ is invertible in $KK^G$. This $KK$-axiom and homotopy-invariance characterize graded $KK^G$-theory universally and completely, thus directly extending the well-known characterization of $KK$-theory for ungraded $C^*$-algebras via stability, homotopy invariance and splitexactness by Higson.

The universal property of graded $KK^G$-theory

Abstract

A universal category-theoretical characterization of groupoid equivariant -theory for -graded -algebras is established, by observing the ``-axiom'' that for each , the `corner-embedding' -homomorphism is invertible in . This -axiom and homotopy-invariance characterize graded -theory universally and completely, thus directly extending the well-known characterization of -theory for ungraded -algebras via stability, homotopy invariance and splitexactness by Higson.
Paper Structure (8 sections, 10 theorems, 42 equations)

This paper contains 8 sections, 10 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. $KK^G$-theory
  3. $GK^G$-theory
  4. The functor ${\mathcal{A}}$
  5. The functor ${\mathcal{B}}$
  6. Connection between ${\mathcal{A}}$ and ${\mathcal{B}}$
  7. Fusion of elements
  8. The standard form

Key Result

Lemma 4.2

Let $J,X$ be ${\mathbb Z}_2$-graded $G$-subalgebras in between as follows: Denote ${\bf J}:=J$ regarded as $G$-equivariant ${\mathbb Z}_2$-graded $X$-module by ordinary algebra multiplication in $X$ ($J \cdot X \subseteq J, \langle J, J \rangle \subseteq J$). Let $s: A \rightarrow X$ (interpreted $X \subseteq {\mathcal{L}}_J(J) \cong {\mathcal{L}}_X({\bf J})$) be a given $ (ii) Assume that ($\i

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2: $GK^G$-theory
  • Definition 4.1: Functor ${\mathcal{A}}$
  • Lemma 4.2
  • proof
  • Corollary 4.3
  • proof
  • Definition 5.1: Functor ${\mathcal{B}}$
  • Lemma 5.2
  • ...and 14 more