Let $l^\infty(\mathbb{N})$ be the $C^*$-algebra of bounded sequences $\mathbb{N}\to\mathbb{C}$.
I would like to compute the $K_1(l^\infty(\mathbb{N}))$, which I suspect is the zero group. However, I'm not sure how to argue this rigorously.
Question: How would one show that $K_1(l^\infty(\mathbb{N}))=0$? (A reference would be great also.)
Thoughts: My first thought was that because $l^\infty(\mathbb{N})=\prod_{\mathbb{N}}\mathbb{C}$, one could use the fact that operator $K$-theory respects direct products. However, for infinite products I think this is only true if each factor in the product has some additional stability property.