It is shown that the ring of fractions of the algebra of all bounded linear operators on a separable infinite dimensional Banach space is isomorphic to the Adams completion of the algebra with respect to a carefully chosen set of morphisms in the category of separable infinite dimensional Banach spaces and bounded linear norm preserving operators of norms at most 1.
"Categorical construction of the ring of fractions." Tbilisi Math. J. 9 (1) 1 - 8, June 2016. https://doi.org/10.1515/tmj-2016-0001