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Discussion

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2029)

Abstract

The discussion undertaken, in Chap. 6, about the functor \(\mathbb{L}\) on regular rings, can be mimicked for the functor \(\mathbb{V}\) (nonstable K-theory) introduced in Example 1.1.3, restricted to (von Neumann) regular rings. It is a fundamental open problem in the theory of regular rings whether every conical refinement monoid, of cardinality at most \(\aleph\)1, is isomorphic to \(\mathbb{L}\)(R) for some regular ring R, cf. Goodearl [25], Ara [3]. Due to counterexamples developed in Wehrung [61], the situation is hopeless in cardinality \(\aleph\)2 or above. Recent advances on those matters, for more general classes of rings such as exchange rings but also for C*-algebras, can be found in Wehrung [71].

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Correspondence to Pierre Gillibert .

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© 2011 Springer-Verlag Berlin Heidelberg

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Gillibert, P., Wehrung, F. (2011). Discussion. In: From Objects to Diagrams for Ranges of Functors. Lecture Notes in Mathematics(), vol 2029. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21774-6_7

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