We study the $K$-theory of the reduced $C^*$-algebra of $GL(n)$ over a local field $F$ with zero characteristic. For the archimedan case we obtain quite explicit formulas and we use automorphic induction to interpret a curious similarity for the $K$-theory of $GL(n,\mathbb{C})$ and $GL(2n,\mathbb{R})$. When $F$ is nonarchimedean we relate base-change with functoriality of affine buildings