H-space structure on the Calkin algebra - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:50:26Z http://mathoverflow.net/feeds/question/69793 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69793/h-space-structure-on-the-calkin-algebra H-space structure on the Calkin algebra Ulrich Pennig 2011-07-08T13:02:40Z 2011-07-08T13:02:40Z <p>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.</p> <blockquote> <p>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).</p> </blockquote>