TITLE: Proof of Bott Periodicity in twisted K-theory
QUESTION [15 upvotes]: 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?

REPLY [10 votes]: Twisted K-theory is just a particular case of K-theory of Banach algebras. Therefore, Bott periodicity is a consequence of general results. See for instance Max Karoubi. Twisted K-theory old and new.