Pseudo-Rank Functions on Rickart *-rings

Abstract

Abstract. Pseudo-rank functions on Rickart *-rings are introduced and their propertiesare studied. 1. IntroductionA real valued function Don a lattice Lis called a dimension function if therange of D has either an upper bound or a lower bound and for all a;b 2L,D(a_b) + D(a^b) = D(a) + D(b), see; von Neumann [12] p.58. The theory ofdimension functions is studied in various structures. von Neumann [12] introduceddimensionality in continuous geometries by using perspectivity, whereas Iwamura[6] used the concept of a relation called the p-relation .Kaplansky [8], Murray and von Neumann [11] and others have introduced dimen-sionality in rings of operators by using equivalence of projections. Maeda [10] gener-alized the work of von Neumann [12] and Kaplansky [8] for a certain class of lattices.At the same time Loomis [9] gave an abstract setting to the Murray, von Neumanndimension theory by using complete orthocomplemented lattices. Berberian [2] hasdeveloped theory of dimension functions on the lattice of projections of a nite Baer*-ring. Goodearl [4] developed the dimension theory for a certain class of modules.von Neumann [12], p.231 has introduced the concept of a rank-function on a regularring which generalizes the dimension function. Goodearl [3], [5] has introduced anddeveloped the study of pseudo-rank functions on regular rings, which is a general-ization of rank functions.In this paper we introduce and study the concept of a pseudo-rank function on aRickart *-ring R. We obtain some basic properties of pseudo-rank functions andthe set of all pseudo-rank functions on R, on the lines of Goodearl [5] for Rickart*-rings. The unde ned terms are from Berberian [2] and Birkho [1].