Liminary C⁎-algebras with boolean spectrum

Abstract

Liminary C⁎-algebras with boolean spectrum are the main topic of Cignoli's joint paper with G.A. Elliott and the present author (Cignoli et al., 1993 [9]). Solving the analogue of Kaplansky's problem for these algebras, in that paper it is proved that the Murray von Neumann order of projections is sufficient to uniquely recover the C⁎-algebraic structure. In this paper we continue the study of these algebras. Among others, we prove that the Elliott partial semigroup of any such algebra A is canonically extendible to a locally finite MV-algebraE(A), in the sense that every finite subset of E(A) generates a finite subalgebra of E(A). Further, every extremal state s of K0(A) has the property that s(K0(A)) is a cyclic subgroup of R, and K0(A) has general comparability. We characterize central projections in these C*-algebras as fixpoints for the “eccentricity” partial order ⊑ on E(A) introduced in the present author's paper (Mundici, 2023 [25]).