Proof of Bott Periodicity in twisted K-theory 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?
 A: I attempted the following baby version. Namely, I asked myself whether there exists a unitary map $S:H\oplus H\rightarrow H$ such that the induced map $Fred(H\oplus H)\rightarrow Fred(H)$ is $U(H)$-equivariant, where $U(H)$ acts in the usual way on $H$ and diagonally on $H\oplus H$. Let $S$ be represented by two operators (not necessarily isomorphisms) $S_i:H\rightarrow H$, so that we can write
$$S=\begin{pmatrix}
    S_1 &   S_2
\end{pmatrix}.
$$
Then, we are asking whether or not we can find $S$ such that $S\circ(T\oplus T)=(T\oplus T)\circ S$ for all unitary opators $T$ on $U$. This forces $T\circ S_1=S_1\circ T$ and $T\circ S_2=S_2\circ T$ for all $T$. This in fact forces $S_i$ to be a unitary isomorphism. Indeed, suppose that $v$ is in the kernel of $S_1$, then $e_1$ is in the kernel of $T\circ S_1$. Let $w$ be an element orthogonal to $v$ and not in the kernel of $S_1$, and let $T$ be the unitary operator that switches $w$ and $v$. Then, $v$ is not in the kernel of $S_1\circ T$. Therefore, $S_i$ has no kernel. The same argument using the adjoints $S_i^*$ show that $S_i$ has no cokernel. Therefore, each $S_i$ is a unitary isomorphism. Then, the commutativity restraint implies that each $S_i$ is diagonal $S_i=\lambda_i Id_{H}$. But, then, $S$ clearly has a non-trivial kernel, which is a contradiction.
Thus, I think that if there is a $PU(H)$-equivariant equivalence as asked in the question, it seems that it cannot come from $PU(H)$ acting diagonally on $S_n\otimes H$ and on $H$.
A: 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.
