I have a question about the Proof of Bott Periodicity in twisted K-theory by Atiyah and Segal in their paper Twisted K-theory.

Following their notation, to prove Bott periodicity in this context it is enough to provide a $U(H)$-equivariant homotopy equivalence $$ Fred^{(0)}(H)\to \Omega^{2}Fred^{0}(H). $$ One may assume that all the spaces in sight have the norm topology for simplicity. This is done in two steps.

Step 1. Take $S_{n}$ an irreducible graded module for the complexified Clifford algebra $C_{n}$. Then for $n$ even, tensoring with $S_{n}$ gives an isomorphism $$ Fred^{0}(H)\to Fred^{n}(S_{n}\otimes H). $$ This map is clearly $U(H)$-equivariant.

Step 2.There is a map $$ Fred^{n}(S_{n}\otimes H)\to \Omega^{n}Fred^{0}(S_{n}\otimes H). $$ which was constructed explicitly by Atiyah and Singer and it is easy to see that it is a $U(H)$-equivariant homotopy equivalence.

However, one would like to get back to $\Omega^{n}Fred^{0}(H)$. The spaces

$$
\Omega^{n}Fred^{0}(S_{n}\otimes H) \text{ and } \Omega^{n}Fred^{0}(H)
$$
are homotopy equivalent but all the maps I seem to be able to construct don't preserve $U(H)$-equivariance and this is taken as granted in the proof by Atiyah and Segal.

Can anyone tell me what I am missing?