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For ${\cal H}$ an infinite-dimensional Hilbert space, the canonical projection $\cal{B(H)}\rightarrow \cal{A(H)}$ (where $\cal{A(H)}$ is the Calkin algebra), has no continuous, linear section; this is due to E.O. Thorp, http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1103038424