Continuity of functors with respect to generalized inductive limits

Abstract

Let ({{\displaystyle A}}_i,|{{\displaystyle \varphi }}_{i,i+\mathrm{1}}) be a generalized inductive system of a sequence (A_i) of unital separable C*-algebras, with A={{\displaystyle \lim }}_{i\to \infty }({{\displaystyle A}}_i,{{\displaystyle \varphi }}_{i,i+\mathrm{1}}). Set {{\displaystyle \varphi }}_{j,i}={{\displaystyle \varphi }}_{i-\mathrm{1},i}\mathrm{o}\cdots \mathrm{o}{{\displaystyle \varphi }}_{j+\mathrm{1},j+\mathrm{2}}\mathrm{o}{{\displaystyle \varphi }}_{j,j+\mathrm{1}} for all i>j: We prove that if {{\displaystyle \varphi }}_{j,i} are order zero completely positive contractions for all j and i>j; and L:=\inf \{\lambda |\lambda \in \sigma ({{\displaystyle \varphi }}_{j,i}({{\displaystyle \mathrm{1}}}_{{{\displaystyle A}}_j}))\} for all j and i>j}>0; where \sigma ({{\displaystyle \varphi }}_{j,i}({{\displaystyle \mathrm{1}}}_{{{\displaystyle A}}_j})) is the spectrum of {{\displaystyle \varphi }}_{j,i}({{\displaystyle \mathrm{1}}}_{{{\displaystyle A}}_j}) ; then {{\displaystyle \lim }}_{i\to \infty }(\mathrm{C}\mathrm{u}({{\displaystyle a}}_i),\mathrm{C}\mathrm{u}({{\displaystyle \varphi }}_{i,i+\mathrm{1}}))=\mathrm{C}\mathrm{u}(A) ; where Cu(A) is a stable version of the Cuntz semigroup of C*-algebra A: Let ({{\displaystyle A}}_n,{{\displaystyle \varphi }}_{m,n}) be a generalized inductive system of C*-algebras, with the {{\displaystyle \varphi }}_{m,n} order zero completely positive contractions. We also prove that if the decomposition rank (nuclear dimension) of A_n is no more than some integer k for each n; then the decomposition rank (nuclear dimension) of A is also no more than k:

Keywords

Completely positive map / order zero / generalized inductive limits / classification of C*-algebra

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