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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).

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I'd like to clarify one point. There exists two H-space structures on U(Q): one for the sum and one for the product of vector bundles. The H-space structure for the sum is easy (and that's not what the question is about). The H-space structure for product is more tricky, and I can't think of an explicit description... –  André Henriques Jul 8 '11 at 18:46
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Thank you, André. Actually, the H-space structure for the sum should be just the composition in the group U(Q), shouldn't it? –  Ulrich Pennig Jul 9 '11 at 9:05
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Yes. You can also take direct sum, followed by the map induced by an isomorphism $H \cong H\oplus H$. –  André Henriques Jul 9 '11 at 11:47

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