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Guillermo Cortiñas

Publications and source records attributed to Guillermo Cortiñas.

At least 19 recordsLinked to original sources
arXiv2025

Homology of Steinberg algebras

We study homological invariants of the Steinberg algebra $\mathcal{A}_k(\mathcal{G})$ of an ample groupoid $\mathcal{G}$ over a commutative ring $k$. For $\mathcal{G}$ principal or Hausdorff with ${\mathcal{G}}^{\rm{Iso}}\setminus{\mathcal{G}}^{(0)}$ discrete, we compute Hochschild and cyclic homology of $\mathcal{A}_k(\mathcal{G})$ in terms of groupoid homology. For any ample Hausdorff groupoid $\mathcal{G}$, we find that $H_*(\mathcal{G})$ is a direct summand of $HH_*(\mathcal{A}_k(\mathcal{G}))$; using this and the Dennis trace we obtain a map $\overline{D}_*:K_*(\mathcal{A}_k(\mathcal{G}))\to H_n(\mathcal{G},k)$. We study this map when $\mathcal{G}$ is the (twisted) Exel-Pardo groupoid associated to a self-similar action of a group $G$ on a graph, and compute $HH_*(\mathcal{A}_k(\mathcal{G}))$ and $H_*(\mathcal{G},k)$ in terms of the homology of $G$, and the $K$-theory of $\mathcal{A}_k(\mathcal{G})$ in terms of that of $k[G]$.

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arXiv2024

Classification conjectures for Leavitt path algebras

The theory of Leavitt path algebras is intrinsically related, via graphs, to the theory of symbolic dynamics and $C^*$-algebras where the major classification programs have been a domain of intense research in the last 50 years. In this survey article, we gather together current lines of research in the classification of Leavitt path algebras, questions, conjectures, and some of the results about them that have been obtained so far.

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arXiv2024

A Quillen model structure of local homotopy equivalences

In this note, we construct a closed model structure on the category of $\mathbb{Z}/2\mathbb{Z}$-graded complexes of projective systems of ind-Banach spaces. When the base field is the fraction field $F$ of a complete discrete valuation ring $V$, the homotopy category of this model structure is the derived category of the quasi-abelian category $\overleftarrow{\mathsf{Ind}(\mathsf{Ban}_F)}$. This homotopy category is the appropriate target of the local and analytic cyclic homology theories for complete, torsionfree $V$-algebras and $\mathbb{F}$-algebras. When the base field is $\mathbb{C}$, the homotopy category is the target of local and analytic cyclic homology for pro-bornological $\mathbb{C}$-algebras, which includes the subcategory of pro-$C^*$-algebras.

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arXiv2023

Exel-Pardo algebras with a twist

Katsura associated a $C^*$-algebra $C^*_{A,B}$ to integral matrices $A\ge 0$ and $B$ of the same size, gave sufficient conditions on $(A,B)$ making it simple purely infinite (SPI), and proved that any separable $C^*$-algebra $KK$-isomorphic to a cone of an element $ξ\in KK(C(S^1)^n,C(S^1)^n)$ in Kasparov's $KK$ is $KK$-isomorphic to an SPI $C^*_{A,B}$. Here we introduce, for the data of a commutative ring $\ell$, matrices $A,B$ as above and $C$ of the same size with coefficients in the group $\mathcal{U}(\ell)$ of invertible elements, an $\ell$-algebra $\mathcal{O}_{A,B}^C$, the twisted Katsura algebra of the triple $(A,B,C)$, show it is SPI whenever $\ell\supset\mathbb{Q}$ is a field and $(A,B)$ satisfy Katsura conditions, and that any $\ell$-algebra which is a cone of a map $ξ\in kk(\ell[t,t^{-1}]^n,\ell[t,t^{-1}]^n)$ in the bivariant algebraic $K$-theory category $kk$ is $kk$-isomorphic to an SPI $\mathcal{O}_{A,B}^C$. Twisted Katsura $\ell$-algebras are twisted Exel-Pardo algebras $L(G,E,ϕ_c)$ associated to a group $G$ acting on a graph $E$, and $1$-cocycles $ϕ:G\times E^1\to G$ and $c:G\times E^1\to \mathcal{U}(\ell)$. We describe $L(G,E,ϕ_c)$ by generators and relations, as a quotient of a twisted semigroup algebra, as a twisted Steinberg algebra, as a corner skew Laurent polynomial algebra, and as a universal localization of a tensor algebra. We use each of these guises of $L(G,E,ϕ_c)$ to study its $K$-theoretic, regularity and simplicity properties. For example we show that if $\ell\supset \mathbb{Q}$ is a field, $G$ and $E$ are countable and $E$ is regular, then $L(G,E,ϕ_c)$ is simple whenever the Exel-Pardo $C^*$-algebra $C^*(G,E,ϕ)$ is, and is SPI if in addition the Leavitt path algebra $L(E)$ is SPI.

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arXiv2023

Simplicity of $L^p$-graph algebras

For each $1\le p<\infty$ and each countable directed graph $E$ we consider the Leavitt path $\mathbb{C}$-algebra $L(E)$ and the $L^p$-operator graph algebra $\mathcal{O}^p(E)$. We show that the (purely infinite) simplicity of $\mathcal{O}^p(E)$ as a Banach algebra is equivalent to the (purely infinite) simplicity of $L(E)$ as a ring.

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arXiv2022

Graded $K$-theory and Leavitt path algebras

Let $G$ be a group and $\ell$ a commutative unital $\ast$-ring with an element $λ\in \ell$ such that $λ+ λ^\ast = 1$. We introduce variants of hermitian bivariant $K$-theory for $\ast$-algebras equipped with a $G$-action or a $G$-grading. For any graph $E$ with finitely many vertices and any weight function $ω\colon E^1 \to G$, a distinguished triangle for $L(E)=L_\ell(E)$ in the hermitian $G$-graded bivariant $K$-theory category $kk^h_{G_{\mathrm{gr}}}$ is obtained, describing $L(E)$ as a cone of a matrix with coefficients in $\mathbb{Z}[G]$ associated to the incidence matrix of $E$ and the weight $ω$. In the particular case of the standard $\mathbb{Z}$-grading, and under mild assumptions on $\ell$, we show that the isomorphism class of $L(E)$ in $kk^h_{\mathbb{Z}_{\mathrm{gr}}}$ is determined by the graded Bowen-Franks module of $E$. We also obtain results for the graded and hermitian graded $K$-theory of $\ast$-algebras in general and Leavitt path algebras in particular which are of independent interest, including hermitian and bivariant versions of Dade's theorem and of Van den Bergh's exact sequence relating graded and ungraded $K$-theory.

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arXiv2021

Lifting graph $C^*$-algebra maps to Leavitt path algebra maps

Let $ξ:C^*(E)\to C^*(F)$ be a unital $*$-homomorphism between simple purely infinite Cuntz-Krieger algebras of finite graphs. We prove that there exists a unital $*$-homomorphism $ϕ:L(E)\to L(F)$ between the corresponding Leavitt path-algebras such that $ξ$ is homotopic to the map $\hatϕ:C^*(E)\to C^*(F)$ induced by completion. We show moreover that $\hatϕ$ is a homotopy equivalence in the $C^*$-algebraic sense if and only if $ϕ$ is a homotopy equivalence in the algebraic, polynomial sense. We deduce, in particular, that any isomorphism between simple purely infinite Cuntz-Krieger algebras is homotopic to the completion of a unital algebraic homotopy equivalence.

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arXiv2021

Nonexistence of graded unital homomorphisms between Leavitt algebras and their Cuntz splices

Let $n\ge 2$, let $\mathcal{R}_n$ be the graph consisting of one vertex and $n$ loops and let $\mathcal{R}_{n^-}$ be its Cuntz splice. Let $L_n=L(\mathcal{R}_n)$ and $L_{n^-}=L(\mathcal{R}_{n^-})$ be the Leavitt path algebras over a unital ring $\ell$. Let $C_m$ be the cyclic group on $2\le m\le \infty$ elements. Equip $L_n$ and $L_{n^-}$ with their natural $C_m$-gradings. We show that under mild conditions on $\ell$, which are satisfied for example when $\ell$ is a field or a PID, there are no unital $C_m$-graded ring homomorphisms $L_n\to L_{n^-}$ nor in the opposite direction.

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arXiv2021

Classifying Leavitt path algebras up to involution preserving homotopy

We prove that the Bowen-Franks group classifies the Leavitt path algebras of purely infinite simple finite graphs over a regular supercoherent commutative ring with involution where $2$ is invertible, equipped with their standard involutions, up to matricial stabilization and involution preserving homotopy equivalence. We also consider a twisting of the standard involution on Leavitt path algebras and obtain partial results in the same direction for purely infinite simple graphs. Our tools are $K$-theoretic, and we prove several results about (Hermitian, bivariant) $K$-theory of Leavitt path algebras.

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arXiv2021

Bivariant Hermitian $K$-theory and Karoubi's fundamental theorem

Let $\ell$ be a commutative ring with involution $*$ containing an element $λ$ such that $λ+λ^*=1$ and let $\operatorname{Alg}^*_\ell$ be the category of $\ell$-algebras equipped with a semilinear involution and involution preserving homomorphisms. We construct a triangulated category $kk^h$ and a functor $j^h:\operatorname{Alg}^*_\ell\to kk^h$ that is homotopy invariant, matricially and hermitian stable and excisive and is universal initial with these properties. We prove that a version of Karoubi's fundamental theorem holds in $kk^h$. By the universal property of the latter, this implies that any functor $H:\operatorname{Alg}^*_\ell\to\mathfrak{T}$ with values in a triangulated category which is homotopy invariant, matricially and hermitian stable and excisive satisfies the fundamental theorem. We also prove a bivariant version of Karoubi's $12$-term exact sequence.

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arXiv2019

Non-Archimedean analytic cyclic homology

Let $V$ be a complete discrete valuation ring with fraction field $F$ of characteristic zero and with residue field $\mathbb{F}$. We introduce analytic cyclic homology of complete torsion-free bornological algebras over $V$. We prove that it is homotopy invariant, stable, invariant under certain nilpotent extensions, and satisfies excision. We use these properties to compute it for tensor products with dagger completions of Leavitt path algebras. If $R$ is a smooth commutative $V$-algebra of relative dimension $1$, then we identify its analytic cyclic homology with Berthelot's rigid cohomology of $R\otimes_{V}\mathbb{F}$.

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arXiv2018

Algebraic bivariant $K$-theory and Leavitt path algebras

This article is the first of two where we investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras $L(E)$ and $L(F)$ of graphs $E$ and $F$ over a commutative ground ring $\ell$. In this first article we consider Leavitt path algebras of general graphs over general ground rings; the second article will focus mostly on purely infinite simple unital Leavitt path algebras over a field. Bivariant algebraic $K$-theory $kk$ is the universal homology theory with the properties above; we prove a structure theorem for unital Leavitt path algebras in $kk$. We show that under very mild assumptions on $\ell$, for a graph $E$ with finitely many vertices and reduced incidence matrix $A_E$, the structure of $L(E)$ depends only on the isomorphism classes of the cokernels of the matrix $I-A_E$ and of its transpose, which are respectively the $kk$ groups $KH^1(L(E))=kk_{-1}(L(E),\ell)$ and $KH_0(L(E))=kk_0(\ell,L(E))$. Hence if $L(E)$ and $L(F)$ are unital Leavitt path algebras such that $KH_0(L(E))\cong KH_0(L(F))$ and $KH^1(L(E))\cong KH^1(L(F))$ then no homology theory with the above properties can distinguish them. We also prove that for Leavitt path algebras, $kk$ has several properties similar to those that Kasparov's bivariant $K$-theory has for $C^*$-graph algebras, including analogues of the Universal coefficient and Künneth theorems of Rosenberg and Schochet.

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arXiv2018

Homotopy classification of Leavitt path algebras

In this paper we address the classification problem for purely infinite simple Leavitt path algebras of finite graphs over a field $\ell$. Each graph $E$ has associated a Leavitt path $\ell$-algebra $L(E)$. There is an open question which asks whether the pair $(K_0(L(E)), [1_{L(E)}])$, consisting of the Grothendieck group together with the class $[1_{L(E)}]$ of the identity, is a complete invariant for the classification, up to algebra isomorphism, of those Leavitt path algebras of finite graphs which are purely infinite simple. We show that $(K_0(L(E)), [1_{L(E)}])$ is a complete invariant for the classification of such algebras up to polynomial homotopy equivalence. To prove this we develop the bivariant algebraic $K$-theory of Leavitt path algebras and obtain several results of independent interest.

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arXiv2018

$L^p$-operator algebras associated with oriented graphs

For each $1\le p<\infty$ and each countable oriented graph $Q$ we introduce an $L^p$-operator algebra $\mathcal{O}^p(Q)$ which contains the Leavitt path $\mathbb{C}$-algebra $L_Q$ as a dense subalgebra and is universal for those $L^p$-representations of $L_Q$ which are spatial in the sense of N.C. Phillips. For $\mathcal{R}_n$ the graph with one vertex and $n$ loops ($2\le n\le \infty$), $\mathcal{O}^p(\mathcal{R}_n)=\mathcal{O}^p_n$, the $L^p$-Cuntz algebra introduced by Phillips. If $p\notin\{1,2\}$ and $\mathcal{S}(Q)$ is the inverse semigroup generated by $Q$, $\mathcal{O}^p(Q)=F_{\operatorname{tight}}^p(\mathcal{S}(Q))$ is the tight semigroup $L^p$-operator algebra introduced by Gardella and Lupini. We prove that $\mathcal{O}^p(Q)$ is simple as an $L^p$-operator algebra if and only if $L_Q$ is simple, and that in this case it is isometrically isomorphic to the closure $\overline{ρ(L_Q)}$ of the image of any nonzero spatial $L^p$-representation $ρ:L_Q\to\mathscr{L}(L^p(X))$. We also show that if $L_Q$ is purely infinite simple and $p\ne p'$, then there is no nonzero continuous homomorphism $\mathcal{O}^p(Q)\to\mathcal{O}^{p'}(Q)$. Our results generalize those obtained by Phillips for $L^p$-Cuntz algebras.

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arXiv2017

Weak completions, bornologies and rigid cohomology

Let $V$ be a complete discrete valuation ring with residue field $k$ of positive characteristic and with fraction field $K$ of characteristic 0. We clarify the analysis behind the Monsky--Washnitzer completion of a commutative $V$-algebra using completions of bornological $V$-algebras. This leads us to a functorial chain complex for a finitely generated commutative algebra over the residue field $k$ that computes its rigid cohomology in the sense of Berthelot.

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arXiv2017

Nonarchimedean bornologies, cyclic homology and rigid cohomology

Let $V$ be a complete discrete valuation ring with residue field $k$ and with fraction field $K$ of characteristic 0. We clarify the analysis behind the Monsky--Washnitzer completion of a commutative $V$-algebra using spectral radius estimates for bounded subsets in complete bornological $V$-algebras. This leads us to a functorial chain complex for commutative $k$-algebras that computes Berthelot's rigid cohomology. This chain complex is related to the periodic cyclic homology of certain complete bornological $V$-algebras.

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arXiv2017

The $K$-theory of toric schemes over regular rings of mixed characteristic

We show that if $X$ is a toric scheme over a regular commutative ring $k$ then the direct limit of the $K$-groups of $X$ taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was previously known for regular commutative rings containing a field. The affine case of our result was conjectured by Gubeladze. We prove analogous results when $k$ is replaced by an appropriate $K$-regular, not necessarily commutative $k$-algebra.

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