# Unitary Cuntz semigroups of ideals and quotients.

@article{Cantier2020UnitaryCS, title={Unitary Cuntz semigroups of ideals and quotients.}, author={Laurent Cantier}, journal={arXiv: Operator Algebras}, year={2020} }

We define a notion of ideals in the category of ordered monoids satisfying the Cuntz axioms introduced in [2] and termed Cu$^\sim$. We show that the set of ideals of a Cu$^\sim$-semigroup $S$ has a complete lattice structure. In fact, we prove that for any separable C*-algebra with stable rank one A, the assignment $\,$ I $\mapsto$ Cu$_1$(I) defines a complete lattice isomorphism between Lat(A) and Lat(Cu$_1$(A)). Further, we introduce the notion of quotient ideals and exactness for the (non… Expand

#### 3 Citations

The unitary Cuntz semigroup on the classification of non-simple C*-algebras

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- 2021

This paper argues that the unitary Cuntz semigroup, introduced in [9] and termed Cu1, contains crucial information regarding the classification of non-simple C-algebras. We exhibit two (non-simple)… Expand

Uniformly PoM-Based Cuntz semigroups and approximate intertwinings

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- 2021

We study topological aspects of the category of abstract Cuntz semigroups, termed Cu. We provide a suitable setting in which we are able to uniformly control how to approach an element of a… Expand

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

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- 2020

We introduce a new invariant for separable C*-algebras of stable rank one that merges the Cuntz semigroup information together with the K$_1$-group information. This semigroup, termed the… Expand

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

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We introduce a new invariant for separable C*-algebras of stable rank one that merges the Cuntz semigroup information together with the K$_1$-group information. This semigroup, termed the… Expand

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This paper argues that the unitary Cuntz semigroup, introduced in [9] and termed Cu1, contains crucial information regarding the classification of non-simple C-algebras. We exhibit two (non-simple)… Expand

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