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Browsing by Author "Kucerovsky, Dan"

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    Cu-nuclearity implies LLP and exactness
    (2021) Ivanescu, Cristian; Kucerovsky, Dan
    The Cu-nuclearity property is an analogue of Skandalis's notion of K-nuclearity, adapted to the case of Cuntz semigroups of C*-algebras. We prove that this implies nuclearity, and we introduce a weaker form of the condition. We prove that the new condition weak Cu-nuclearity, for simple separable C*-algebras, implies exactness and the local lifting property (LLP). We also prove that if A is a simple C*-algebra with the weak Cu-nuclearity property, and B is any simple C*-algebra, then A ⊗min B = A ⊗max B. We prove that Cu-nuclearity does imply nuclearity, and that in some cases this is also true for weak Cu-nuclearity.
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    Traces and Pedersen ideals of tensor products of non-unital C*-algebras
    (2019) Ivanescu, Cristian; Kucerovsky, Dan
    We show that positive elements of a Pedersen ideal of a tensor product can be approximated in a particularly strong sense by sums of tensor products of positive elements. This has a range of applications to the structure of tracial cones and related topics, such as the Cuntz-Pedersen space or the Cuntz semigroup. For example, we determine the cone of lower semicontinuous traces of a tensor product in terms of the traces of the tensor factors, in an arbitrary C*-tensor norm. We show that the positive elements of a Pedersen ideal are sometimes stable under Cuntz equivalence. We generalize a result of Pedersen’s by showing that certain classes of completely positive maps take a Pedersen ideal into a Pedersen ideal. We provide theorems that in many cases compute the Cuntz semigroup of a tensor product.
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    Villadsen idempotents
    (2024) Ivanescu, Cristian; Kucerovsky, Dan; Markin, Marat V.; Nikolaev, Igor V.; Trunk, Carsten
    C*-algebras are rings, sometimes nonunital, obeying certain axioms that ensure a very well-behaved representation theory upon Hilbert space. Moreover, there are some wellknown features of the representation theory leading to subtle questions about norms on tensor products of C*-algebras, and thus to the subclass of nuclear C*-algebras. The question whether all separable nuclear C*-algebras satisfy the Universal Coefficient Theorem (UCT) remains one of the most important open problems in the structure and classification theory of such algebras. One of the most promising ways to test the UCT conjecture depends on finding C*- algebras that behave as idempotents under the tensor product, and satisfy certain additional properties. Briefly put, if there exists a simple, separable, and nuclear C*-algebra that is an idempotent under the tensor product, satisfies a certain technical property, and is not one of the already known such elements {𝑂∞,𝑂2,UHF∞, 𝐽,𝑍,ℂ,} then the UCT fails. Although we do not disprove the UCT in this publication, we do find new idempotents in the class of Villadsen algebras.