Some remarks in $C^{\ast }$- and $K$-theory

Abstract

This note consists of three unrelated remarks. First, we demonstrate how roughly speaking $\ast$-homomorphisms between matrix stable $C^{\ast }$-algebras are exactly the uniformly continuous $\ast$-preserving group homomorphisms between their genral linear groups. Second, using the Cuntz picture in $KK$-theory we bring morphisms in $KK$-theory represented by generators and relations to a particular simple form. Third, we show that for an inverse semigroup its associated groupoid is Hausdorff if and only if the inverse semigroup is $E$-continuous.