# Group of invertible isometries on $\ell^p$

I know that the group of unitary operators on infinite dimensional separable Hilbert space is connected but I would like to know whether the group of invertible isometries on $$\ell^p$$ is connected when $$p\in[1,\infty)\setminus\{ 2 \}$$. I haven't been able to get an answer (I don't know whether the Banach-Lamperti theorem can be used in any way) and I also can't seem to find relevant literature so I am looking for help here.

• Careful: in finite dimensions, the group of unitary operators on a Hilbert space is disconnected, with one connected component being the special unitary operators (with determinant $1$) and the other connected component being the unitary operators with determinant $-1$. While the determinant may not be $\pm 1$ in the $\ell^p$ case, I still think you'll get a positive/negative determinant split in finite dimensions. I don't know how this generalises to infinite dimensions, but since your question doesn't explicitly exclude finite dimensions, I think some clarification would be good. – Theo Bendit Oct 11 '18 at 12:42
• @TheoBendit : Thanks for spotting this. I did mean the infinite dimensional case. – cyc Oct 11 '18 at 14:13
• Obviously I don't know the answer here, but could I ask, why is the group of unitary operators on infinite-dimensional separable Hilbert spaces necessarily connected? Do you where this is proven? – Theo Bendit Oct 12 '18 at 2:25
• You can look at Kuiper's paper "The homotopy type of the unitary group of Hilbert space" where it is shown that the unitary group of separable infinite dimensional Hilbert space is contractible to a point. – cyc Oct 12 '18 at 13:20