TITLE: What is the group of automorphisms of $l^{\infty}$?
QUESTION [5 upvotes]: What is the group of automorphisms of $l^{\infty}$?
I think it would be the permutations of the integers. Is this right?

REPLY [6 votes]: This is to record what is probably the easiest proof (that the answer is "yes"), following a remark by Nik Weaver.
Noting that $\ell^\infty(\mathbb{Z})\simeq C(\beta\mathbb{Z})$, using Banach-Stone, we see that every auto gives a homeo of $\beta\mathbb{Z}$.
But since $\mathbb{Z}$ is the subset of isolated points, every homeo preseves it, and since it is dense the homeo is determined by its restriction to $\mathbb{Z}$.