JOURNAL OF ALGEBRA 110, 56-73 (1987) The Jordan Regular Ring Associated to a Finite J BW-Algebra PEDRO JIMENEZ GARIJO Departamento de Algebra y Fundamentos, Colegro Umversltario de Almeria, Universidad de Granada, Almeria, Spain AND ANGEL RODRIGUEZ PALACIOS Departamento de Teoriu de Funciones, Facuitod de Ciencius, Unirersidad de Grrmada, Granada, Sparn Communicated br N. Jacobson Received July 1, 1984 If A is a finite von Neumann algebra, then there exists a *-regular ring R (in the sense of von Neumann) whose lattice of principal left ideals is isomorphic to the lattice of projections of A (it is said that R coordinatizes A); this ring was constructed by Murray and von Neumann [14] by enlarging A to contain certain unbounded operators defined on dense linear subspacesof the Hilbert space on which A acts. By using an abstract version of the Murray-von Neumann construction, Berberian showed in [4] that a finite A W*-algebra A is always contained in a continuous *-regular ring R such that R has no new projections. Later, Hafner [7] and Pyle [ 161 showed that the regular ring constructed by Berberian is the maximal ring of quotients of A. The analogous problem for Rickart *-rings was considered by Han- delman [9], who constructs, for a finite Rickart C*-algebra A, a *-regular ring R containing A such that R has no new projections. Later, Ara and Menal [3] showed that the regular ring constructed by Handelman is the classical ring of quotients of A. In the case of Jordan algebras, Ayupov [l] has given an enlargement of a JW-algebra, similar to the one of Murray and von Neumann for W*-algebras, without going into the problem of regularity or ring of quotients in the finite case. 56 0021~8693/87 $3.00 Copynghl 0 1987 by Academic Press, Inc All rights of reproduction m any form reserved