A Simple Criterion for Extending Natural Transformations to Higher $K$-Theory

In this article we introduce a very simple an widely applicable criterion for extending natural transformations to higher $K$-theory. More precisely, we prove that every natural transformation defined on the Grothendieck group and with values in an additive theory admits a unique extension to higher $K$-theory. As an application, the higher trace maps and the higher Chern characters originally constructed by Dennis and Karoubi, respectively, can be obtained in an elegant, unified, and conceptual way from our general results.

2010 Mathematics Subject Classification: 19D55, 18G55

Keywords and Phrases: Higher $K$-theory, higher trace maps, higher Chern characters, non-commutative motives

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