Injectivity of the Connecting Maps in AH Inductive Limit Systems

Abstract

AbstractLetAbe the inductive limit of a systemwith, whereXn,iis a finite simplicial complex, andPn,iis a projection inM[n,i](C(Xn,i)). In this paper, we will prove thatAcan be written as another inductive limitwith, whereYn,iis a finite simplicial complex, andQn,iis a projection inM{n,i}(C(Yn,i)), with the extra condition that all the mapsψn,n+1areinjective. (The result is trivial if one allows the spacesYn,ito be arbitrary compact metrizable spaces.) This result is important for the classification of simple AH algebras. The special case that the spacesXn,iare graphs is due to the third author.