Quantitative K-theory for geometric operator algebras
The purpose of this chapter is to give a friendly introduction to quantitative Ktheory of operator algebras and its applications. Quantitative operator K-theory was first introduced in my work on the Novikov conjecture for groups with finite asymptotic dimension [Yu98]. Hervé Oyono-Oyono and I developed a more general quantitative K-theory for C *-algebras [OOY15]. Quantitative operator theory provides a constructive way to compute K-theory of C *-algebras under certain finiteness conditions. The crucial idea is that quantitative operator K-theory is often computable by using a cutting-and-pasting technique in each scale under certain finite-dimensionality conditions and the usual K-theory is an inductive limit when the scale goes to infinity.
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