Already in Utumi’s original work in the 1950’s a symmetric version of his famous maximal ring of quotients appeared but was not pursued further. Later Schelter and Lanning took up his ideas and began a systematic study. More recently, Ortega computed the maximal symmetric ring of quotients for path algebras and found an alternative approach via bicategories.

In an ongoing research
project with Pere Ara
( Barcelona
) we are studying an
analytic companion, the maximal C*-algebra of quotients. In this talk I shall
report on some first insights of this work in progress, in particular in
comparison to the local multiplier algebra of a C*-algebra and with special
attention to the case of AW*-algebras.