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arXiv · 2012.04338

A unitary Cuntz semigroup for C*-algebras of stable rank one

Laurent Cantier
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Abstract

We introduce a new invariant for C*-algebras of stable rank one that merges the Cuntz semigroup information together with the K$_1$-group information. This semigroup, termed the Cu$_1$-semigroup, is constructed as equivalence classes of pairs consisting of a positive element in the stabilization of the given C*-algebra together with a unitary element of the unitization of the hereditary subalgebra generated by the given positive element. We show that the Cu$_1$-semigroup is a well-defined continuous functor from the category of C*-algebras of stable rank one to a suitable codomain category that we write Cu$^\sim$. Furthermore, we compute the Cu$_1$-semigroup of some specific classes of C*-algebras. Finally, in the course of our investigation, we show that we can recover functorially Cu, K$_1$ and K$_*:=$K$_0$$\oplus$ K$_1$ from Cu$_1$.

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BibTeXRIS

Laurent Cantier. 2020-12-08. A unitary Cuntz semigroup for C*-algebras of stable rank one. https://doi.org/10.1016/j.jfa.2021.109175

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