# Quantitative *K*-theory for geometric operator algebras

## Abstract

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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