Ring structure on K-theory modeled on fredholm operators

Question

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.

Answer 1

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