In non-commutative Iwasawa theory K-theoretic properties of Iwasawa algebras, i.e. completed group rings of e.g. compact p-adic Lie groups play a crucial role. Such groups arise naturally as Galois groups attached to p-adic representations as for example on the Tate module of abelian varieties. In this talk we address in particular the question for which such groups the invariant $SK_1$ vanishes. We reduce this vanishing to a linear algebra problem for Lie algebras over arbitrary rings, which we solve for Chevalley orders in split reductive Lie algebras. Also we shall try to indicate what arithmetic consequences the vanishing of $SK_1$ has.