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.

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