Computing the $K$-homology $K$-theory product in splitexact algebraic $KK$-theory

TL;DR

This paper provides explicit formulas for computing products in split-exact algebraic KK-theory and establishes a functor linking algebraic KK-theory to $kk$-theory, enhancing computational accessibility.

Contribution

It introduces explicit formulas for KK-theory products and verifies a functor from algebraic split-exact KK-theory to $kk$-theory, bridging the two frameworks.

Findings

Explicit formulas for KK-theory products with special G-actions

Verification of a functor from algebraic KK-theory to $kk$-theory

Enhanced computational methods for KK-theory products

Abstract

Explicit formulas are indicated that compute the product $z \cdot w$ of a level-one element $z \in KK^G(A,{\bf C})$ and any element $w \in KK^G({\bf C},B)$ in splitexact algebraic $KK^G$-theory, or $KK^G$-theory for $C^*$-algebras, with very special $G$-actions. We also make such products accessible to linear-split half-exact $kk$-theory by verifying the existence of a functor from algebraic splitexact $KK$-theory to $kk$-theory.