A note on the K\"unneth theorem for nonnuclear C*-algebras

TL;DR

This paper explores the K"unneth theorem in K-theory for nonnuclear C*-algebras, highlighting differences between minimal and maximal tensor products through specific examples.

Contribution

It provides a detailed analysis of when the K"unneth theorem holds for minimal versus maximal tensor products in nonnuclear C*-algebras, using illustrative examples.

Findings

Some algebras satisfy the K"unneth theorem for minimal tensor products but not for maximal ones.

Other algebras satisfy the theorem for maximal tensor products but not for minimal ones.

Examples by Skandalis demonstrate these distinctions.

Abstract

In this mostly expository note, we revisit the K\"unneth theorem in $K$-theory of nonnuclear C*-algebras. We show that, using examples considered by Skandalis, there are algebras satisfying the K\"unneth theorem for the minimal tensor product but not for the maximal tensor product and vice versa.