# Some remarks in $C^*$- and $K$-theory

## TL;DR

This paper presents three distinct results in $C^*$- and $K$-theory, including characterizations of homomorphisms, simplifications in $KK$-theory, and conditions for the Hausdorff property of groupoids associated with inverse semigroups.

## Contribution

It provides new insights into the structure of $C^*$-algebras, $KK$-theory representations, and the topology of inverse semigroup groupoids.

## Key findings

- $*$-homomorphisms correspond to uniformly continuous group homomorphisms
- Simplified form of $KK$-theory morphisms using generators and relations
- Hausdorff condition for inverse semigroup groupoids characterized by $E$-continuity

## Abstract

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 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.