By the Atiyah-Jänich theorem the K-group $K^0(X)$ for a compact space $X$ may be represented as $[X, U(Q)]$, where $Q = B(H)/K(H)$ is the Calkin algebra and $H$ is a separable infinite dimensional Hilbert space. But $K^0(X)$ is a ring with multiplication induced by the tensor product of vector bundles. This should induce an $H$-space structure on the unitaries of the Calkin algebra, which should come from the "tensor product" of Fredholm operators.
Is there an explicit description of this $H$-space structure on $U(Q)$? That is: Can you give the multiplication map $U(Q) \times U(Q) \to U(Q)$ explicitely? (hmm, this still is vague, but I hope you understand what I mean).