This paper is intended as a sequel to "Finite Rickart C* algebras and their properties" (Studies in Analysis, Adu. in Math. 4 (1980), 171-19.5), referred to throughout as I. The referencing here is exactly the same as that of I, and the sections are numbered as a continuation of I.
In I, for every finite Rickart C* algebra, there was constructed a von Neumann regular overring and this was exploited to prove right projections are equivalent to left projections, as well as related results. Here we further employ the regular overring to obtain new results. For instance, if the finite Rickart C* algebra is either monotone u-complete or is already an II x n matrix over some other ring (for some n > l), then all matrix rings over it are Rickart, finite, and satisfy polar decomposition. A more striking result is that such rings are actually classical orders in their regular rings (that is. the nonzero-divisors satisfy the Ore condition and their inverses generate the regular ring). This is new even for finite W* factors.
We also widen the class of Rickart C* algebras satisfying the equivalence of right and left projections to include (non-finite) monotone u-complete C* algebras.
In Section 6, we prove preliminary results on monotone a-complete C* algebras. In Section 7, these are made to yield the matrix ring results. Section 8 deals with orders.
The following is a synopsis of definitions and results from I.
A C* algebra T is Rickart, if for all a in T, there exists a projectionp in T such that (t E T 1 at = 0) (denoted a + or a') is (1 -p) T, p is referred to as the right projection of T (RP(t) =p). The left projection is defined similarly (LP). A Rickart C* algebra is finite if xx* = 1 implies x*x = 1.
In case the Rickart C* algebra T has the property that LP(t) is equivalent to RP(t) for all t in T, then we say T has the property LP -RP (see [2]).
THEOREM 2.1 of I. Given a finite Rickart C* algebra T, there exists a von Neumann regular overring R such that (1) the involution on R extends to R so that for all r in R, there exists a projection p with rR =pR; (2)