So, if we have an infinite dimensional Hilbert space $H$ then the way you put a ring structure on $F(H)$ is by taking the isomorphism $H\oplus H \to H$ we can define the sum of two Fredholm operators as 
$$ H \to H \oplus H \to H \oplus H \to H$$
where the middle map is the sum of the two operators.


What is the equivalent for the ring structure? I figured it's something to do with
$$ H \to H \otimes H \oplus H \otimes H \to H \otimes H \oplus H \otimes H \to H$$
but I cannot get the signs on the image map to work out. 
Also, is there a good reference to all this? Especially on how this works out when we switch to spectra.



Ring structure on K-theory modeled on fredholm operators
The formula is $$A \cdot B = \begin{bmatrix}
A \otimes I & -I \otimes B^\* \\
I \otimes B & A^\* \otimes I 
\end{bmatrix}$$
the sign $- I \otimes B^\*$ is to make associativity work out. 


I found it in here: Klaus Janich. Vektorraumbundel und der Raum der Fredholm-Operatoren. Math. Ann., 161:129–142, 1965


