Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 10 / 4 / d / Solution

Let $c=(1-ab{)}^{-1}\in A$. From the two inverse identities we have $cab=c-1=abc$. The candidate inverse isIndeedandBoth products have been checked, so this is valid in a noncommutative Banach algebra. It is the identity underlying the nonzero spectra of products in opposite orders result.