An element of a Banach algebra An element of a Banach algebra which has no logarithm.

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Consider the unilateral shift operator u on a separable Hilbert space H, then u is Fredholm of index nul u - def u = 0-1 = -1. If p:B(H)® [ B(H)/ K(H)] is the quotient map and p(u) = ew for some w in the Calkin algebra [ B(H)/ K(H)], then there exists an element w¢ Î B(H) with p(w¢) = w, so p(u) = ew = ep(w¢) = p(ew¢). Hence u-ew¢ Î K(H). But ew¢ is invertible and so ind u = ind(ew¢) = 0, a contradiction.


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On 22 Feb 2001, 00:13.