Projections in free product C $^*$ -algebras, II

A classification result is obtained for the C * -algebras that are (stably isomorphic to) inductive limits of 1-dimensional noncommutative CW complexes with trivial K1-group. The classifying functor Cu ∼ is defined in terms of the Cuntz semigroup of the unitization of the algebra. For the simple C * -algebras covered by the classification, Cu ∼ reduces to the ordered K0-group, the cone of traces, and the pairing between them. As an application of the classification, it is shown that the crossed products by a quasi-free action O2 ⋊ λ R are all isomorphic for a dense set of positive irrational numbers λ.Theorem 1.0.1. Let A be either a 1-dimensional noncommutative CW complex with trivial K 1 -group, or a sequential inductive limit of such C * -algebras, or a C * -algebra stably isomorphic to one such inductive limit. Let B be a stable rank one C * -algebra. Then for every morphism in the category Cu α : Cu ∼ (A) → Cu ∼ (B)

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Classification of inductive limits of 1-dimensional NCCW complexes

Robert

2012

Advances in Mathematics

101

225

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Closed convex hulls of unitary orbits in C⁎-algebras of real rank zero

Skoufranis

2016

Journal of Functional Analysis

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In this paper, we study closed convex hulls of unitary orbits in various C * -algebras. For unital C * -algebras with real rank zero and a faithful tracial state determining equivalence of projections, a notion of majorization describes the closed convex hulls of unitary orbits for self-adjoint operators. Other notions of majorization are examined in these C * -algebras. Combining these ideas with the Dixmier property, we demonstrate unital, infinite dimensional C * -algebras of real rank zero and strict comparison of projections with respect to a faithful tracial state must be simple and have a unique tracial state. Also, closed convex hulls of unitary orbits of self-adjoint operators are fully described in unital, simple, purely infinite C * -algebras.

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Optimal transport and unitary orbits in C⁎-algebras

Jacelon

Strung

Vignati

2021

Journal of Functional Analysis

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Projections in free product C $^*$ -algebras, II

Cited by 6 publications

References 26 publications

Classification of inductive limits of 1-dimensional NCCW complexes

Classification of inductive limits of 1-dimensional NCCW complexes

Closed convex hulls of unitary orbits in C⁎-algebras of real rank zero

Optimal transport and unitary orbits in C⁎-algebras

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