I will present a unified approach to several constructions in operator
algebras that are related to the algebraic theory of quotient rings, showing deep
relations between algebraic and analytic concepts. These constructions include
the algebra of unbounded operators affiliated to a finite von Neumann algebra,
defined by Murray and von Neumann in 1936, and the C*-algebra of essential multipliers
of a C*-algebra, defined by Elliott in 1976. In both cases, the algebras were
defined without any explicit reference to localisation theory, but the fact
that they can be obtained as quotient algebras has played a very important role in
several applications.

Created: 4/7/01 Updated: 4/7/01