Some remarks in C^*- and K-theory

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