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Luis Santiago

Publications and source records attributed to Luis Santiago.

14 recordsLinked to original sources
arXiv2020

Rokhlin dimension: duality, tracial properties, and crossed products

We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of C*-algebras, including: $D$-absorbing C*-algebras, where $D$ is a strongly self-absorbing C*-algebra, stable C*-algebras, C*-algebras with finite nuclear dimension (or decomposition rank), C*-algebras with finite stable rank (or real rank), and C*-algebras whose K-theory is either trivial, rational, or $n$-divisible for $n\in\mathbb{N}$. The combination of nuclearity and the UCT is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous $C(X)$-algebra with fibers that are stably isomorphic to the underlying algebra. The space $X$ is computed in some cases of interest, and we use its description to construct a $\mathbb{Z}_2$-action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.

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arXiv2020

Decomposition rank of approximately subhomogeneous C*-algebras

It is shown that every Jiang-Su stable approximately subhomogeneous C*-algebra has finite decomposition rank. Previously, it was not even known that such algebras have finite nuclear dimension. A key step in the proof is that subhomogeneous C*-algebra are locally approximated by a certain class of more tractable subhomogeneous algebras, namely, a non-commutative generalization of the class of cell complexes. The result is applied to show that Jiang-Su stable minimal Z-crossed products have finite decomposition rank.

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arXiv2019

A revised augmented Cuntz semigroup

We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of 1-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: We show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups.

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arXiv2016

The equivariant Cuntz semigroup

We introduce an equivariant version of the Cuntz semigroup, that takes an action of a compact group into account. The equivariant Cuntz semigroup is naturally a semimodule over the representation semiring of the given group. Moreover, this semimodule satisfies a number of additional structural properties. We show that the equivariant Cuntz semigroup, as a functor, is continuous and stable. Moreover, cocycle conjugate actions have isomorphic associated equivariant Cuntz semigroups. One of our main results is an analog of Julg's theorem: the equivariant Cuntz semigroup is canonically isomorphic to the Cuntz semigroup of the crossed product. We compute the induced semimodule structure on the crossed product, which in the abelian case is given by the dual action. As an application of our results, we show that freeness of a compact Lie group action on a compact Hausdorff space can be characterized in terms of a canonically defined map into the equivariant Cuntz semigroup, extending results of Atiyah and Segal for equivariant $K$-theory. Finally, we use the equivariant Cuntz semigroup to classify locally representable actions on direct limits of one-dimensional NCCW-complexes, generalizing work of Handelman and Rossmann.

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arXiv2015

Equivariant *-homomorphisms, Rokhlin constraints and equivariant UHF-absorption

We classify equivariant *-homomorphisms between C*-dynamical systems associated to actions of finite groups on C*-algebras with the Rokhlin property. In addition, the given actions are classified. An obstruction is obtained for the Cuntz semigroup of a C*-algebra allowing such an action. We also obtain an equivariant UHF-absorption result.

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arXiv2014

Crossed product by actions of finite groups with the Rokhlin Property

We introduce and study a Rokhlin-type property for actions of finite groups on (not necessarily unital) C*-algebras. We show that the corresponding crossed product C*-algebras can be locally approximated by C*-algebras that are stably isomorphic to closed two-sided ideals of the given algebra. We then use this result to prove that several classes of C*-algebras are closed under crossed products by finite group actions with this Rokhlin property.

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arXiv2012

Reduction of the dimension of nuclear C*-algebras

We show that for a large class of C*-algebras $\mathcal{A}$, containing arbitrary direct limits of separable type I C*-algebras, the following statement holds: If $A\in \mathcal{A}$ and $B$ is a simple projectionless C*-algebra with trivial K-groups that can be written as a direct limit of a system of (nonunital) recursive subhomogeneous algebras with no dimension growth then the stable rank of $A\otimes B$ is one. As a consequence we show that if $A\in \mathcal A$ then the stable rank of $A\otimes\mathcal W$ is one. We also prove the following stronger result: If $A$ is separable C*-algebra that can be written as a direct limit of C*-algebras of the form $\mathrm{C}_0(X)\otimes \mathrm{M}_n$, where $X$ is locally compact and Hausdorff, then $A\otimes \mathcal W$ can be written as a direct limit of a sequence of 1-dimensional noncommutative CW-complexes.

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arXiv2011

Recovering the Elliott invariant from the Cuntz semigroup

Let $A$ be a simple, separable C$^*$-algebra of stable rank one. We prove that the Cuntz semigroup of $\CC(\T,A)$ is determined by its Murray-von Neumann semigroup of projections and a certain semigroup of lower semicontinuous functions (with values in the Cuntz semigroup of $A$). This result has two consequences. First, specializing to the case that $A$ is simple, finite, separable and $\mathcal Z$-stable, this yields a description of the Cuntz semigroup of $\CC(\T,A)$ in terms of the Elliott invariant of $A$. Second, suitably interpreted, it shows that the Elliott functor and the functor defined by the Cuntz semigroup of the tensor product with the algebra of continuous functions on the circle are naturally equivalent.

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arXiv2011

Pullbacks, $C(X)$-algebras, and their Cuntz semigroup

In this paper we analyse the structure of the Cuntz semigroup of certain $C(X)$-algebras, for compact spaces of low dimension, that have no $\mathrm{K}_1$-obstruction in their fibres in a strong sense. The techniques developed yield computations of the Cuntz semigroup of some surjective pullbacks of C$^*$-algebras. As a consequence, this allows us to give a complete description, in terms of semigroup valued lower semicontinuous functions, of the Cuntz semigroup of $C(X,A)$, where $A$ is a not necessarily simple C$^*$-algebra of stable rank one and vanishing $\mathrm{K}_1$ for each closed, two sided ideal. We apply our results to study a variety of examples.

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arXiv2009

Classification of homomorphisms from $C_0(0,1]$ to a C*-algebra

A class of C*-algebras is described for which the homomorphism from $C_0(0,1]$ to the algebra may be classified by means of the Cuntz semigroup functor. Examples are given of algebras--simple and non-simple--for which this classification fails. It is shown that a suitable suspension of the Cuntz semigroup functor deals successfully with some of these counterexamples.

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arXiv2009

The cone of lower semicontinuous traces on a C*-algebra

The cone of lower semicontinuous traces is studied with a view to its use as an invariant. Its properties include compactness, Hausdorffness, and continuity with respect to inductive limits. A suitable notion of dual cone is given. The cone of lower semicontinuous 2-quasitraces on a (non-exact) C*-algebra is considered as well. These results are applied to the study of the Cuntz semigroup. It is shown that if a C*-algebra absorbs the Jiang-Su algebra, then the subsemigroup of its Cuntz semigroup consisting of the purely non-compact elements is isomorphic to the dual cone of the cone of lower semicontinuous 2-quasitraces. This yields a computation of the Cuntz semigroup for the following two classes of C*-algebras: C*-algebras that absorb the Jiang-Su algebra and have no non-zero simple subquotients, and simple C*-algebras that absorb the Jiang-Su algebra.

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arXiv2007

Cuntz semigroups of ideals and quotients and a generalized Kasparov Stabilization Theorem

Let A be a C*-algebra and I a closed two-sided ideal of A. We use the Hilbert C*-modules picture of the Cuntz semigroup to investigate the relations between the Cuntz semigroups of I, A and A/I. We obtain a relation on two elements of the Cuntz semigroup of A that characterizes when they are equal in the Cuntz semigroup of A/I. As a corollary, we show that the Cuntz semigroup functor is exact. Replacing the Cuntz equivalence relation of Hilbert modules by their isomorphism, we obtain a generalization of Kasparov's Stabilization theorem.

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