Let $\kappa$ be an uncountable cardinal. What is the K-theory for the C*-algebras $\mathcal{K}(\ell_2(\kappa))$ and $\mathcal{B}(\ell_2(\kappa))$, of, respectively, compact and bounded operators on $\ell_2(\kappa)$?
K-theory for non-separable C*-algebras
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$\begingroup$
operator-algebras
c-star-algebras
topological-k-theory
k-theory
1 Answers
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For every Hilbert space $H$, $K_0(\mathcal{K}(H))=\mathbb{Z}$ because the only compact projections in $\mathcal{B}(H^{\oplus n})\cong M_n(\mathcal{B}(H))$ are finite-rank projections and these are distinguished only by dimension. Actually $K_0(\mathcal{K}(E)) = \mathbb{Z}$ for any Banach space $E$. Moreover, if $H$ is infinite-dimensional then $K_0(\mathcal{B}(H))=\{0\},$ because $H\cong H\oplus H$, hence $[{\rm id}_H]\oplus [{\rm id}_H] = [{\rm id}_H]$ so it must be the trivial group.
As Atkinson's theorem works well in non-separable spaces, the $K_1$-groups are the same as for the separable Hilbert space.