Resumen de Injectivity of the connecting maps in AH inductive limit systems

Liangqing Li, George A. Elliott, Guihua Gong

• Let A be the inductive limit of a system A1 xrightarrow(phi1,2) A2 xrightarrow(phi2,3) A3 longrightarrow ... with An = bigoplusi=1tn Pn,i M[n,i](C(Xn,i))Pn,i, where Xn,i is a finite simplicial complex, and Pn,i is a projection in M[n,i](C(Xn,i)). In this paper, we will prove that A can be written as another inductive limit B1 xrightarrow(psi1,2) B2 xrightarrow(psi2,3) B3 longrightarrow ... with Bn = bigoplusi=1sn Qn,i M{n,i}(C(Yn,i)) Qn,i, where Yn,i is a finite simplicial complex, and Qn,i is a projection in M{n,i} (C(Yn,i)), with the extra condition that all the maps psin,n+1 are injective. (The result is trivial if one allows the spaces Yn,i to be arbitrary compact metrizable spaces.) This result is important for the classification of simple AH algebras. The special case that the spaces Xn,i are graphs is due to the third named author.