Injectivity of the Connecting Maps in AH Inductive Limit Systems

Canadian mathematical bulletin

关键词: Yang-Mills connection;    vector bundle;    gauge transformation;   

DOI  :  10.4153/CMB-2005-005-9

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $A$ be the inductive limit of a system$$A_{1}xrightarrow{phi_{1,2}}A_{2}xrightarrow{phi_{2,3}} A_{3}longrightarrow cd$$with $A_n =igoplus_{i=1}^{t_n} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$, where$~X_{n,i}$ is a finite simplicial complex, and $P_{n,i}$ is aprojection in $M_{[n,i]}(C(X_{n,i}))$. In this paper, we willprove that $A$ can be written as another inductive limit$$B_1xrightarrow{psi_{1,2}} B_2xrightarrow{psi_{2,3}} B_3longrightarrow cd $$with $B_n =igoplus_{i=1}^{s_n} Q_{n,i}M_{{n,i}}(C(Y_{n,i}))Q_{n,i}$,where $Y_{n,i}$ is a finite simplicial complex, and $Q_{n,i}$ is aprojection in $M_{{n,i}}(C(Y_{n,i}))$, with the extra conditionthat all the maps $psi_{n,n+1}$ are emph{injective}. (Theresult is trivial if one allows the spaces $Y_{n,i}$ to bearbitrary compact metrizable spaces.) This result is importantfor the classification of simple AH algebras (seecite{G5,G6,EGL}. The special case that the spaces $X_{n,i}$ aregraphs is due to the third named author cite{Li1}.

【 授权许可】

Unknown