In Progress
Bosa, Joan; Perera, Francesc; Wu, Jianchao; Zacharias, Joachim
Continuous type semigroups, dynamical strict comparison, and ideal-free quotients of Cuntz semigroups Unpublished
0001.
@unpublished{BosPerWuZac2b,
title = {Continuous type semigroups, dynamical strict comparison, and ideal-free quotients of Cuntz semigroups},
author = {Joan Bosa and Francesc Perera and Jianchao Wu and Joachim Zacharias},
year = {0001},
date = {0001-00-06},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
Bosa, Joan; Perera, Francesc; Wu, Jianchao; Zacharias, Joachim
Almost elementary dynamical systems and dynamical strict comparison Unpublished
0001.
@unpublished{BosPerWuZac,
title = {Almost elementary dynamical systems and dynamical strict comparison },
author = { Joan Bosa and Francesc Perera and Jianchao Wu and Joachim Zacharias},
year = {0001},
date = {0001-00-05},
journal = {In Progress},
pages = {25},
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pubstate = {published},
tppubtype = {unpublished}
}
Preprints
Antoine, Ramon; Perera, Francesc; Robert, Leonel; Thiel, Hannes
Traces on ultrapowers of C*-algebras Unpublished
0002.
@unpublished{AntPerRobThi:traces,
title = {Traces on ultrapowers of C*-algebras},
author = {Ramon Antoine and Francesc Perera and Leonel Robert and Hannes Thiel},
year = {0002},
date = {0002-00-00},
journal = {In Progress},
pages = {22},
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pubstate = {published},
tppubtype = {unpublished}
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Forthcoming
Antoine, Ramon; Perera, Francesc; Robert, Leonel; Thiel, Hannes
Traces on ultrapowers of C*-algebras Unpublished
0002.
@unpublished{AntPerRobThi:traces,
title = {Traces on ultrapowers of C*-algebras},
author = {Ramon Antoine and Francesc Perera and Leonel Robert and Hannes Thiel},
year = {0002},
date = {0002-00-00},
journal = {In Progress},
pages = {22},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
Publications
2022
Antoine, Ramon; Perera, Francesc; Robert, Leonel; Thiel, Hannes
C*-algebras of stable rank one and their Cuntz semigroups Journal Article
In: Duke Mathematical Journal, vol. 171, iss. 1, pp. 33–99, 2022.
@article{AntPerRobThi:sr1,
title = {C*-algebras of stable rank one and their Cuntz semigroups },
author = {Ramon Antoine and Francesc Perera and Leonel Robert and Hannes Thiel },
doi = {10.1215/00127094-2021-0009},
year = {2022},
date = {2022-02-01},
urldate = {2022-02-01},
journal = {Duke Mathematical Journal},
volume = {171},
issue = {1},
pages = {33--99},
abstract = {The uncovering of new structure on the Cuntz semigroup of a C*-algebra of stable rank one leads to several applications: We answer affirmatively, for the class of stable rank one C*-algebras, a conjecture by Blackadar and Handelman on dimension functions, the Global Glimm Halving problem, and the problem of realizing functions on the cone of 2-quasitraces as ranks of Cuntz semigroup elements. We also gain new insights into the comparability properties of positive elements in C*-algebras of stable rank one. },
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The uncovering of new structure on the Cuntz semigroup of a C*-algebra of stable rank one leads to several applications: We answer affirmatively, for the class of stable rank one C*-algebras, a conjecture by Blackadar and Handelman on dimension functions, the Global Glimm Halving problem, and the problem of realizing functions on the cone of 2-quasitraces as ranks of Cuntz semigroup elements. We also gain new insights into the comparability properties of positive elements in C*-algebras of stable rank one.
2021
Antoine, Ramon; Perera, Francesc; Robert, Leonel; Thiel, Hannes
Edwards’ condition for Quasitraces on C*-algebras Journal Article
In: Proceedings of the Royal Society of Edinburgh, Section A, vol. 151, no. 2, pp. 525-547, 2021.
@article{AntPerRobThi:Edw,
title = {Edwards’ condition for Quasitraces on C*-algebras},
author = {Ramon Antoine and Francesc Perera and Leonel Robert and Hannes Thiel },
doi = {10.1017/prm.2020.26},
year = {2021},
date = {2021-04-01},
journal = {Proceedings of the Royal Society of Edinburgh, Section A},
volume = {151},
number = {2},
pages = {525-547},
abstract = {We prove that Cuntz semigroups of C*-algebras satisfy Edwards' condition with respect to every quasitrace. This condition is a key ingredient in the study of the realization problem of functions on the cone of quasitraces as ranks of positive elements. In the course of our investigation, we identify additional structure of the Cuntz semigroup of an arbitrary C*-algebra and of the cone of quasitraces. },
keywords = {},
pubstate = {published},
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}
We prove that Cuntz semigroups of C*-algebras satisfy Edwards' condition with respect to every quasitrace. This condition is a key ingredient in the study of the realization problem of functions on the cone of quasitraces as ranks of positive elements. In the course of our investigation, we identify additional structure of the Cuntz semigroup of an arbitrary C*-algebra and of the cone of quasitraces.
2020
Antoine, Ramon; Perera, Francesc; Thiel, Hannes
Cuntz semigroups of ultraproduct C*-algebras Journal Article
In: Journal of the London Mathematical Society, vol. 102, no. 3, pp. 994-1029, 2020.
@article{AntPerThi:ultra,
title = {Cuntz semigroups of ultraproduct C*-algebras },
author = {Ramon Antoine and Francesc Perera and Hannes Thiel},
doi = {10.1112/jlms.12343},
year = {2020},
date = {2020-12-01},
journal = {Journal of the London Mathematical Society},
volume = {102},
number = {3},
pages = {994-1029},
abstract = {We prove that the category of abstract Cuntz semigroups is bicomplete. As a consequence, the category admits products and ultraproducts. We further show that the scaled Cuntz semigroup of the (ultra)product of a family of C*-algebras agrees with the (ultra)product of the scaled Cuntz semigroups of the involved C*-algebras.
As applications of our results, we compute the non-stable K-Theory of general (ultra)products of C*-algebras and we characterize when ultraproducts are simple. We also give criteria that determine order properties of these objects, such as almost unperforation. },
keywords = {},
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We prove that the category of abstract Cuntz semigroups is bicomplete. As a consequence, the category admits products and ultraproducts. We further show that the scaled Cuntz semigroup of the (ultra)product of a family of C*-algebras agrees with the (ultra)product of the scaled Cuntz semigroups of the involved C*-algebras.
As applications of our results, we compute the non-stable K-Theory of general (ultra)products of C*-algebras and we characterize when ultraproducts are simple. We also give criteria that determine order properties of these objects, such as almost unperforation.
As applications of our results, we compute the non-stable K-Theory of general (ultra)products of C*-algebras and we characterize when ultraproducts are simple. We also give criteria that determine order properties of these objects, such as almost unperforation.
Antoine, Ramon; Perera, Francesc; Thiel, Hannes
Abstract bivariant Cuntz semigroups Journal Article
In: International Mathematics Research Notices, vol. 2020, no. 17, pp. 5342-5386, 2020.
@article{AntPerThi:Biv,
title = {Abstract bivariant Cuntz semigroups},
author = {Ramon Antoine and Francesc Perera and Hannes Thiel},
doi = {10.1093/imrn/rny143},
year = {2020},
date = {2020-09-01},
journal = {International Mathematics Research Notices},
volume = {2020},
number = {17},
pages = {5342-5386},
abstract = {We show that abstract Cuntz semigroups form a closed symmetric monoidal category. Thus, given Cuntz semigroups $S$ and $T$, there is another Cuntz semigroup $[[S,T]]$ playing the role of morphisms from $S$ to $T$. Applied to C*-algebras $A$ and $B$, the semigroup $[[mathrm{Cu}(A),mathrm{Cu}(B)]]$ should be considered as the target in analogues of the UCT for bivariant theories of Cuntz semigroups. Abstract bivariant Cuntz semigroups are computable in a number of interesting cases. We also show that order-zero maps between C*-algebras naturally define elements in the respective bivariant Cuntz semigroup. },
keywords = {},
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We show that abstract Cuntz semigroups form a closed symmetric monoidal category. Thus, given Cuntz semigroups $S$ and $T$, there is another Cuntz semigroup $[[S,T]]$ playing the role of morphisms from $S$ to $T$. Applied to C*-algebras $A$ and $B$, the semigroup $[[mathrm{Cu}(A),mathrm{Cu}(B)]]$ should be considered as the target in analogues of the UCT for bivariant theories of Cuntz semigroups. Abstract bivariant Cuntz semigroups are computable in a number of interesting cases. We also show that order-zero maps between C*-algebras naturally define elements in the respective bivariant Cuntz semigroup.
Antoine, Ramon; Perera, Francesc; Thiel, Hannes
Abstract bivariant Cuntz semigroups II Journal Article
In: Forum Mathematicum, vol. 32, no. 1, pp. 45-62, 2020.
@article{AntPerThi:Biv2,
title = {Abstract bivariant Cuntz semigroups II},
author = {Ramon Antoine and Francesc Perera and Hannes Thiel},
doi = {10.1515/forum-2018-0285},
year = {2020},
date = {2020-01-01},
journal = {Forum Mathematicum},
volume = {32},
number = {1},
pages = {45-62},
abstract = {We previously showed that abstract Cuntz semigroups form a closed symmetric monoidal category. This automatically provides additional structure in the category, such as a composition and an external tensor product, for which we give concrete constructions in order to be used in applications.
We further analyse the structure of not necessarily commutative Cu-semi-rings and we obtain, under mild conditions, a new characterization of solid Cu-semirings $R$ by the condition that $Rcong [![ R,R ]!]$. },
keywords = {},
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We previously showed that abstract Cuntz semigroups form a closed symmetric monoidal category. This automatically provides additional structure in the category, such as a composition and an external tensor product, for which we give concrete constructions in order to be used in applications.
We further analyse the structure of not necessarily commutative Cu-semi-rings and we obtain, under mild conditions, a new characterization of solid Cu-semirings $R$ by the condition that $Rcong [![ R,R ]!]$.
We further analyse the structure of not necessarily commutative Cu-semi-rings and we obtain, under mild conditions, a new characterization of solid Cu-semirings $R$ by the condition that $Rcong [![ R,R ]!]$.
2018
Antoine, Ramon; Perera, Francesc; Petzka, Henning
Perforation conditions and almost algebraic order in Cuntz semigroups Journal Article
In: Proceedings of the Royal Society of Edinburgh, Section A, vol. 148, pp. 669-702, 2018.
@article{AntPerPet,
title = {Perforation conditions and almost algebraic order in Cuntz semigroups},
author = {Ramon Antoine and Francesc Perera and Henning Petzka},
doi = {10.1017/S0308210518000069},
year = {2018},
date = {2018-08-01},
journal = {Proceedings of the Royal Society of Edinburgh, Section A},
volume = {148},
pages = {669-702},
abstract = {For a C*-algebra $A$, it is an important problem to determine the Cuntz semigroup $mathrm{Cu}(Aotimesmathcal{Z})$ in terms of $mathrm{Cu}(A)$. We approach this problem from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups, by analysing the passage of significant properties from $mathrm{Cu}(A)$ to $mathrm{Cu}(A)otimes_mathrm{Cu}mathrm{Cu}(mathcal{Z})$. We describe the effect of the natural map $mathrm{Cu}(A)tomathrm{Cu}(A)otimes_mathrm{Cu}mathrm{Cu}(mathcal{Z})$ in the order of $mathrm{Cu}(A)$, and show that, if $A$ has real rank zero and no elementary subquotients, $mathrm{Cu}(A)otimes_mathrm{Cu}mathrm{Cu}(mathcal{Z})$ enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, nonelementary, real rank zero and stable rank one situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with $mathcal{Z}$ is inert at the level of the Cuntz semigroup. },
keywords = {},
pubstate = {published},
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}
For a C*-algebra $A$, it is an important problem to determine the Cuntz semigroup $mathrm{Cu}(Aotimesmathcal{Z})$ in terms of $mathrm{Cu}(A)$. We approach this problem from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups, by analysing the passage of significant properties from $mathrm{Cu}(A)$ to $mathrm{Cu}(A)otimes_mathrm{Cu}mathrm{Cu}(mathcal{Z})$. We describe the effect of the natural map $mathrm{Cu}(A)tomathrm{Cu}(A)otimes_mathrm{Cu}mathrm{Cu}(mathcal{Z})$ in the order of $mathrm{Cu}(A)$, and show that, if $A$ has real rank zero and no elementary subquotients, $mathrm{Cu}(A)otimes_mathrm{Cu}mathrm{Cu}(mathcal{Z})$ enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, nonelementary, real rank zero and stable rank one situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with $mathcal{Z}$ is inert at the level of the Cuntz semigroup.
Antoine, Ramon; Perera, Francesc; Thiel, Hannes
Tensor products and regularity properties of Cuntz semigroups Journal Article
In: Memoirs of the American Mathematical Society, vol. 251, no. 1199, pp. 199 pp., 2018.
@article{AntPerThi:Tensor,
title = {Tensor products and regularity properties of Cuntz semigroups },
author = {Ramon Antoine and Francesc Perera and Hannes Thiel},
doi = { 10.1090/memo/1199},
year = {2018},
date = {2018-01-01},
journal = {Memoirs of the American Mathematical Society},
volume = {251},
number = {1199},
pages = {199 pp.},
abstract = {The Cuntz semigroup of a C*-algebra is an important invariant in the structure and classification theory of C*-algebras. It captures more information than K-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups.
Given a C*-algebra $A$, its (concrete) Cuntz semigroup $Cu(A)$ is an object in the category $Cu$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter $Cu$-semigroups.
We establish the existence of tensor products in the category $Cu$ and study the basic properties of this construction. We show that $Cu$ is a symmetric, monoidal category and relate $Cu(Aotimes B)$ with $Cu(A)otimes_{Cu}Cu(B)$ for certain classes of C*-algebras.
As a main tool for our approach we introduce the category $W$ of pre-completed Cuntz semigroups. We show that $Cu$ is a full, reflective subcategory of $W$. One can then easily deduce properties of $Cu$ from respective properties of $W$, e.g. the existence of tensor products and inductive limits. The advantage is that constructions in $W$ are much easier since the objects are purely algebraic.
We also develop a theory of $Cu$-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing C*-algebra has a natural product giving it the structure of a $Cu$-semiring. We give explicit characterizations of $Cu$-semimodules over such $Cu$-semirings. For instance, we show that a $Cu$-semigroup $S$ tensorially absorbs the $Cu$-semiring of the Jiang-Su algebra if and only if $S$ is almost unperforated and almost divisible, thus establishing a semigroup version of the Toms-Winter conjecture. },
keywords = {},
pubstate = {published},
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}
The Cuntz semigroup of a C*-algebra is an important invariant in the structure and classification theory of C*-algebras. It captures more information than K-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups.
Given a C*-algebra $A$, its (concrete) Cuntz semigroup $Cu(A)$ is an object in the category $Cu$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter $Cu$-semigroups.
We establish the existence of tensor products in the category $Cu$ and study the basic properties of this construction. We show that $Cu$ is a symmetric, monoidal category and relate $Cu(Aotimes B)$ with $Cu(A)otimes_{Cu}Cu(B)$ for certain classes of C*-algebras.
As a main tool for our approach we introduce the category $W$ of pre-completed Cuntz semigroups. We show that $Cu$ is a full, reflective subcategory of $W$. One can then easily deduce properties of $Cu$ from respective properties of $W$, e.g. the existence of tensor products and inductive limits. The advantage is that constructions in $W$ are much easier since the objects are purely algebraic.
We also develop a theory of $Cu$-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing C*-algebra has a natural product giving it the structure of a $Cu$-semiring. We give explicit characterizations of $Cu$-semimodules over such $Cu$-semirings. For instance, we show that a $Cu$-semigroup $S$ tensorially absorbs the $Cu$-semiring of the Jiang-Su algebra if and only if $S$ is almost unperforated and almost divisible, thus establishing a semigroup version of the Toms-Winter conjecture.
Given a C*-algebra $A$, its (concrete) Cuntz semigroup $Cu(A)$ is an object in the category $Cu$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter $Cu$-semigroups.
We establish the existence of tensor products in the category $Cu$ and study the basic properties of this construction. We show that $Cu$ is a symmetric, monoidal category and relate $Cu(Aotimes B)$ with $Cu(A)otimes_{Cu}Cu(B)$ for certain classes of C*-algebras.
As a main tool for our approach we introduce the category $W$ of pre-completed Cuntz semigroups. We show that $Cu$ is a full, reflective subcategory of $W$. One can then easily deduce properties of $Cu$ from respective properties of $W$, e.g. the existence of tensor products and inductive limits. The advantage is that constructions in $W$ are much easier since the objects are purely algebraic.
We also develop a theory of $Cu$-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing C*-algebra has a natural product giving it the structure of a $Cu$-semiring. We give explicit characterizations of $Cu$-semimodules over such $Cu$-semirings. For instance, we show that a $Cu$-semigroup $S$ tensorially absorbs the $Cu$-semiring of the Jiang-Su algebra if and only if $S$ is almost unperforated and almost divisible, thus establishing a semigroup version of the Toms-Winter conjecture.
2014
Perera, Francesc; Toms, Andrew S.; White, Stuart; Winter, Wilhelm
The Cuntz semigroup and stability of close C*-algebras Journal Article
In: Analysis and PDE, vol. 7, pp. 929-952, 2014.
@article{PerTomWhiWin,
title = {The Cuntz semigroup and stability of close C*-algebras},
author = {Francesc Perera and Andrew S. Toms and Stuart White and Wilhelm Winter},
doi = {10.2140/apde.2014.7.929},
year = {2014},
date = {2014-07-01},
journal = {Analysis and PDE},
volume = {7},
pages = {929-952},
abstract = {We prove that separable C*-algebras which are completely close in a natural uniform sense have isomorphic Cuntz semigroups, continuing a line of research developed by Kadison - Kastler, Christensen, and Khoshkam. This result has several applications: we are able to prove that the property of stability is preserved by close C*-algebras provided that one algebra has stable rank one; close C*-algebras must have affinely homeomorphic spaces of lower-semicontinuous quasitraces; strict comparison is preserved by sufficient closeness of C*-algebras. We also examine C*-algebras which have a positive answer to Kadison's Similarity Problem, as these algebras are completely close whenever they are close. A sample consequence is that sufficiently close C*-algebras have isomorphic Cuntz semigroups when one algebra absorbs the Jiang-Su algebra tensorially.
},
keywords = {},
pubstate = {published},
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We prove that separable C*-algebras which are completely close in a natural uniform sense have isomorphic Cuntz semigroups, continuing a line of research developed by Kadison - Kastler, Christensen, and Khoshkam. This result has several applications: we are able to prove that the property of stability is preserved by close C*-algebras provided that one algebra has stable rank one; close C*-algebras must have affinely homeomorphic spaces of lower-semicontinuous quasitraces; strict comparison is preserved by sufficient closeness of C*-algebras. We also examine C*-algebras which have a positive answer to Kadison's Similarity Problem, as these algebras are completely close whenever they are close. A sample consequence is that sufficiently close C*-algebras have isomorphic Cuntz semigroups when one algebra absorbs the Jiang-Su algebra tensorially.
Antoine, Ramon; Dadarlat, Marius; Perera, Francesc; Santiago, Luis
Recovering the Elliott invariant from the Cuntz semigroup Journal Article
In: Transactions of the American Mathematical Society, vol. 366, no. 6, pp. 2907-2922, 2014.
@article{AntDadPerSan,
title = {Recovering the Elliott invariant from the Cuntz semigroup},
author = {Ramon Antoine and Marius Dadarlat and Francesc Perera and Luis Santiago},
doi = {10.1090/S0002-9947-2014-05833-9},
year = {2014},
date = {2014-02-16},
journal = {Transactions of the American Mathematical Society},
volume = {366},
number = {6},
pages = {2907-2922},
abstract = {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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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.
Antoine, Ramon; Bosa, Joan; Perera, Francesc; Petzka, Henning
Geometric structure of dimension functions of certain continuous fields Journal Article
In: Journal of Functional Analysis, vol. 266, no. 4, pp. 2403-2423, 2014.
@article{AntBosPerPet,
title = {Geometric structure of dimension functions of certain continuous fields},
author = {Ramon Antoine and Joan Bosa and Francesc Perera and Henning Petzka},
doi = {10.1016/j.jfa.2013.09.013},
year = {2014},
date = {2014-02-15},
urldate = {0001-00-00},
journal = {Journal of Functional Analysis},
volume = {266},
number = {4},
pages = {2403-2423},
abstract = {In this paper we study structural properties of the Cuntz semigroup and its functionals for continuous fields of C*-algebras over finite dimensional spaces. In a variety of cases, this leads to an answer to a conjecture posed by Blackadar and Handelman. Enroute to our results, we determine when the stable rank of continuous fields of C*-algebras over one dimensional spaces is one. },
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In this paper we study structural properties of the Cuntz semigroup and its functionals for continuous fields of C*-algebras over finite dimensional spaces. In a variety of cases, this leads to an answer to a conjecture posed by Blackadar and Handelman. Enroute to our results, we determine when the stable rank of continuous fields of C*-algebras over one dimensional spaces is one.
2013
Antoine, Ramon; Bosa, Joan; Perera, Francesc
The Cuntz semigroup of continuous fields Journal Article
In: Indiana University Mathematics Journal, vol. 62, no. 4, pp. 1105-1131, 2013.
@article{AntBosPer,
title = {The Cuntz semigroup of continuous fields},
author = {Ramon Antoine and Joan Bosa and Francesc Perera},
doi = {10.1512/iumj.2013.62.5071},
year = {2013},
date = {2013-10-01},
journal = {Indiana University Mathematics Journal},
volume = {62},
number = {4},
pages = {1105-1131},
abstract = {In this paper we describe the Cuntz semigroup of continuous fields of C*-algebras over one dimensional spaces whose fibers have stable rank one and trivial $K_1$ for each closed, two-sided ideal. This is done in terms of the semigroup of global sections on a certain topological space built out of the Cuntz semigroups of the fibers of the continuous field. When the fibers have furthermore real rank zero, and taking into account the action of the space, our description yields that the Cuntz semigroup is a classifying invariant if and only if so is the sheaf induced by the Murray-von Neumann semigroup. },
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In this paper we describe the Cuntz semigroup of continuous fields of C*-algebras over one dimensional spaces whose fibers have stable rank one and trivial $K_1$ for each closed, two-sided ideal. This is done in terms of the semigroup of global sections on a certain topological space built out of the Cuntz semigroups of the fibers of the continuous field. When the fibers have furthermore real rank zero, and taking into account the action of the space, our description yields that the Cuntz semigroup is a classifying invariant if and only if so is the sheaf induced by the Murray-von Neumann semigroup.
Pasnicu, Cornel; Perera, Francesc
The Cuntz semigroup, a Riesz type interpolation property, comparison and the ideal property Journal Article
In: Publicacions Matemàtiques, vol. 57, no. 2, pp. 359-377, 2013.
@article{PasPer,
title = {The Cuntz semigroup, a Riesz type interpolation property, comparison and the ideal property},
author = {Cornel Pasnicu and Francesc Perera},
doi = {10.5565/PUBLMAT 57213 04},
year = {2013},
date = {2013-05-01},
journal = {Publicacions Matemàtiques},
volume = {57},
number = {2},
pages = {359-377},
abstract = {We define a Riesz type interpolation property for the Cuntz semigroup of a C*-algebra and prove it is satisfied by the Cuntz semigroup of every C*-algebra with the ideal property. Related to this, we obtain two characterizations of the ideal property in terms of the Cuntz semigroup of the C*-algebra. Some additional characterizations are proved in the special case of the stable, purely infinite $C^*$-algebras, and two of them are expressed in language of the Cuntz semigroup. We introduce a notion of comparison of positive elements for every unital C*-algebra that has (normalized) quasitraces. We prove that large classes of C*-algebras (including large classes of AH algebras) with the ideal property have this comparison property.},
keywords = {},
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}
We define a Riesz type interpolation property for the Cuntz semigroup of a C*-algebra and prove it is satisfied by the Cuntz semigroup of every C*-algebra with the ideal property. Related to this, we obtain two characterizations of the ideal property in terms of the Cuntz semigroup of the C*-algebra. Some additional characterizations are proved in the special case of the stable, purely infinite $C^*$-algebras, and two of them are expressed in language of the Cuntz semigroup. We introduce a notion of comparison of positive elements for every unital C*-algebra that has (normalized) quasitraces. We prove that large classes of C*-algebras (including large classes of AH algebras) with the ideal property have this comparison property.
2011
Ortega, Eduard; Perera, Francesc; Rørdam, Mikael
The corona factorization property, stability, and the Cuntz semigroup of a C*-algebra Journal Article
In: International Mathematics Research Notices, no. 1, pp. 34-66, 2011.
@article{OrtPerRor:cfp,
title = {The corona factorization property, stability, and the Cuntz semigroup of a C*-algebra},
author = {Eduard Ortega and Francesc Perera and Mikael Rørdam},
doi = {10.1093/imrn/rnr013},
year = {2011},
date = {2011-10-01},
journal = {International Mathematics Research Notices},
number = {1},
pages = {34-66},
abstract = {The Corona Factorization Property, originally invented to study extensions of C*-algebras, conveys essential information about the intrinsic structure of the C*-algebras. We show that the Corona Factorization Property of a $sigma$-unital C*-algebra is completely captured by its Cuntz semigroup (of equivalence classes of positive elements in the stabilization of A). The corresponding condition in the Cuntz semigroup is a very weak comparability property termed the Corona Factorization Property for semigroups. Using this result one can for example show that all unital C*-algebras with finite decomposition rank have the Corona Factorization Property.
Applying similar techniques we study the related question of when C*-algebras are stable. We give an intrinsic characterization, that we term property (S), of C*-algebras that have no non-zero unital quotients and no non-zero bounded 2-quasitraces. We then show that property (S) is equivalent to stability provided that the Cuntz semigroup of the C*-algebras satisfies another (also very weak) comparability property, that we call the $omega$-comparison property. },
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The Corona Factorization Property, originally invented to study extensions of C*-algebras, conveys essential information about the intrinsic structure of the C*-algebras. We show that the Corona Factorization Property of a $sigma$-unital C*-algebra is completely captured by its Cuntz semigroup (of equivalence classes of positive elements in the stabilization of A). The corresponding condition in the Cuntz semigroup is a very weak comparability property termed the Corona Factorization Property for semigroups. Using this result one can for example show that all unital C*-algebras with finite decomposition rank have the Corona Factorization Property.
Applying similar techniques we study the related question of when C*-algebras are stable. We give an intrinsic characterization, that we term property (S), of C*-algebras that have no non-zero unital quotients and no non-zero bounded 2-quasitraces. We then show that property (S) is equivalent to stability provided that the Cuntz semigroup of the C*-algebras satisfies another (also very weak) comparability property, that we call the $omega$-comparison property.
Applying similar techniques we study the related question of when C*-algebras are stable. We give an intrinsic characterization, that we term property (S), of C*-algebras that have no non-zero unital quotients and no non-zero bounded 2-quasitraces. We then show that property (S) is equivalent to stability provided that the Cuntz semigroup of the C*-algebras satisfies another (also very weak) comparability property, that we call the $omega$-comparison property.
Antoine, Ramon; Bosa, Joan; Perera, Francesc
Completions of monoids with applications to the Cuntz semigroup Journal Article
In: International Journal of Mathematics, vol. 22, no. 6, pp. 837-861, 2011.
@article{AntBosPer:completion,
title = {Completions of monoids with applications to the Cuntz semigroup},
author = {Ramon Antoine and Joan Bosa and Francesc Perera},
doi = {10.1142/S0129167X11007057},
year = {2011},
date = {2011-09-01},
journal = {International Journal of Mathematics},
volume = {22},
number = {6},
pages = {837-861},
abstract = {We provide an abstract categorical framework that relates the Cuntz semigroups of the C*-algebras $A$ and $Aotimes mathcal{K}$. This is done through a certain completion of ordered monoids by adding suprema of countable ascending sequences. Our construction is rather explicit and we show it is functorial and unique up to isomorphism. This approach is used in some applications to compute the stabilized Cuntz semigroup of certain C*-algebras.},
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We provide an abstract categorical framework that relates the Cuntz semigroups of the C*-algebras $A$ and $Aotimes mathcal{K}$. This is done through a certain completion of ordered monoids by adding suprema of countable ascending sequences. Our construction is rather explicit and we show it is functorial and unique up to isomorphism. This approach is used in some applications to compute the stabilized Cuntz semigroup of certain C*-algebras.
Ortega, Eduard; Perera, Francesc; Rørdam, Mikael
The corona factorization property and refinement monoids Journal Article
In: Transactions of the American Mathematical Society, vol. 363, no. 9, pp. 4505-4525, 2011.
@article{OrtPerRor:cfpref,
title = {The corona factorization property and refinement monoids},
author = {Eduard Ortega and Francesc Perera and Mikael Rørdam},
doi = {10.1090/S0002-9947-2011-05480-2},
year = {2011},
date = {2011-06-01},
journal = {Transactions of the American Mathematical Society},
volume = {363},
number = {9},
pages = {4505-4525},
abstract = {The Corona Factorization Property of a C*-algebra, originally defined to study extensions of C*-algebras, has turned out to say something important about intrinsic structural properties of the C*-algebra. We show in this paper that a $sigma$-unital C*-algebra A of real rank zero has the Corona Factorization roperty if and only if its monoid V(A) of Murray-von Neumann equivalence classes of projections in matrix algebras over A has a certain (rather weak) comparability property that we call the Corona Factorization Property (for monoids). We show that a projection in such a C*-algebra is properly infinite if (and only if) a multiple of it is properly infinite.
The latter result is obtained from some more general result we establish about conical refinement monoids. We show that the set of order units (together with the zero-element) in a conical refinement monoid is again a refinement monoid under the assumption that the monoid satisfies weak divisibility; and if u is an element in a refinement monoid such that nu is properly infinite, then u can be written as a sum u = s + t such that ns and nt are properly infinite. },
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The Corona Factorization Property of a C*-algebra, originally defined to study extensions of C*-algebras, has turned out to say something important about intrinsic structural properties of the C*-algebra. We show in this paper that a $sigma$-unital C*-algebra A of real rank zero has the Corona Factorization roperty if and only if its monoid V(A) of Murray-von Neumann equivalence classes of projections in matrix algebras over A has a certain (rather weak) comparability property that we call the Corona Factorization Property (for monoids). We show that a projection in such a C*-algebra is properly infinite if (and only if) a multiple of it is properly infinite.
The latter result is obtained from some more general result we establish about conical refinement monoids. We show that the set of order units (together with the zero-element) in a conical refinement monoid is again a refinement monoid under the assumption that the monoid satisfies weak divisibility; and if u is an element in a refinement monoid such that nu is properly infinite, then u can be written as a sum u = s + t such that ns and nt are properly infinite.
The latter result is obtained from some more general result we establish about conical refinement monoids. We show that the set of order units (together with the zero-element) in a conical refinement monoid is again a refinement monoid under the assumption that the monoid satisfies weak divisibility; and if u is an element in a refinement monoid such that nu is properly infinite, then u can be written as a sum u = s + t such that ns and nt are properly infinite.
Antoine, Ramon; Perera, Francesc; Santiago, Luis
Pullbacks, C(X)-algebras, and their Cuntz semigroup Journal Article
In: Journal of Functional Analysis, vol. 260, no. 10, pp. 2844-2880, 2011.
@article{AntPerSan,
title = {Pullbacks, C(X)-algebras, and their Cuntz semigroup},
author = {Ramon Antoine and Francesc Perera and Luis Santiago},
doi = {10.1016/j.jfa.2011.02.016},
year = {2011},
date = {2011-04-01},
journal = {Journal of Functional Analysis},
volume = {260},
number = {10},
pages = {2844-2880},
abstract = {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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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.
Ara, Pere; Perera, Francesc; Toms, Andrew S.
K-theory for operator algebras. Classification of C*-algebras Book Chapter
In: Ara, Pere; Lledó, Fernando; Perera, Francesc (Ed.): Aspects of operator algebras and applications, Contemporary Mathematics, vol. 534, Chapter 1, pp. 1-71, American Mathematical Society, Providence, RI, 2011.
@inbook{AraPerTom,
title = {K-theory for operator algebras. Classification of C*-algebras},
author = {Pere Ara and Francesc Perera and Andrew S. Toms},
editor = {Pere Ara and Fernando Lledó and Francesc Perera},
doi = {10.1090/conm/534/10521},
year = {2011},
date = {2011-02-01},
booktitle = {Aspects of operator algebras and applications, Contemporary Mathematics},
journal = {Contemporary Mathematics},
volume = {534},
pages = {1-71},
publisher = {American Mathematical Society},
address = {Providence, RI},
chapter = {1},
abstract = {In this article we survey some of the recent goings-on in the classification programme of C*-algebras, following the interesting link found between the Cuntz semigroup and the classical Elliott invariant and the fact that the Elliott conjecture does not hold at its boldest. We review the construction of this object both by means of positive elements and via its recent interpretation using countably generated Hilbert modules (due to Coward, Elliott and Ivanescu). The passage from one picture to another is presented with full, concise, proofs. We indicate the potential role of the Cuntz semigroup in future classification results, particularly for non-simple algebras. },
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In this article we survey some of the recent goings-on in the classification programme of C*-algebras, following the interesting link found between the Cuntz semigroup and the classical Elliott invariant and the fact that the Elliott conjecture does not hold at its boldest. We review the construction of this object both by means of positive elements and via its recent interpretation using countably generated Hilbert modules (due to Coward, Elliott and Ivanescu). The passage from one picture to another is presented with full, concise, proofs. We indicate the potential role of the Cuntz semigroup in future classification results, particularly for non-simple algebras.
Ara, Pere; Lledó, Fernando; Perera, Francesc
Appendix: basic definitions and results for operator algebras Book Chapter
In: Ara, Pere; Lledó, Fernando; Perera, Francesc (Ed.): Aspects of operator algebras and applications, Contemporary Mathematics, vol. 534, pp. 157-168, American Mathematical Society, Providence, RI, 2011.
@inbook{AraLlePer,
title = {Appendix: basic definitions and results for operator algebras},
author = {Pere Ara and Fernando Lledó and Francesc Perera},
editor = {Pere Ara and Fernando Lledó and Francesc Perera},
doi = {10.1090/conm/534/10525},
year = {2011},
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booktitle = {Aspects of operator algebras and applications, Contemporary Mathematics},
volume = {534},
pages = {157-168},
publisher = {American Mathematical Society},
address = {Providence, RI},
abstract = {We collect some standard definitions, results and examples of the theory of C*-algebras and von Neumann algebras.},
type = {Appendix},
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pubstate = {published},
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We collect some standard definitions, results and examples of the theory of C*-algebras and von Neumann algebras.
Kucerovsky, Dan; Perera, Francesc
Purely infinite corona algebras of simple C*-algebras with real rank zero Journal Article
In: Journal of Operator Theory, vol. 65, no. 1, pp. 131-144, 2011.
@article{KucPer,
title = {Purely infinite corona algebras of simple C*-algebras with real rank zero},
author = {Dan Kucerovsky and Francesc Perera},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/picoronajot.pdf},
year = {2011},
date = {2011-01-20},
journal = {Journal of Operator Theory},
volume = {65},
number = {1},
pages = {131-144},
abstract = {We explore conditions on simple non-unital C*-algebras with real rank zero and stable rank one under which their corona algebras are purely infinite and not necessarily simple. In particular, our results allow us to characterize when the corona algebra of a simple AF-algebra is purely infinite in terms of continuity conditions on its scale.},
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We explore conditions on simple non-unital C*-algebras with real rank zero and stable rank one under which their corona algebras are purely infinite and not necessarily simple. In particular, our results allow us to characterize when the corona algebra of a simple AF-algebra is purely infinite in terms of continuity conditions on its scale.
2010
Aranda-Pino, Gonzalo; Goodearl, Ken R.; Perera, Francesc; Siles-Molina, Mercedes
Non-simple purely infinite rings Journal Article
In: American Journal of Mathematics, vol. 132, no. 3, pp. 563-610, 2010.
@article{AraGooPerSil,
title = {Non-simple purely infinite rings },
author = {Gonzalo Aranda-Pino and Ken R. Goodearl and Francesc Perera and Mercedes Siles-Molina},
doi = {10.1353/ajm.0.0119},
year = {2010},
date = {2010-12-14},
journal = {American Journal of Mathematics},
volume = {132},
number = {3},
pages = {563-610},
abstract = {In this paper we introduce the concept of purely infinite rings, which in the simple case agrees with the already existing notion of pure infiniteness. We establish various permanence properties of this notion, with respect to passage to matrix rings, corners, and behaviour under extensions, so being purely infinite is preserved under Morita equivalence. We show that a wealth of examples falls into this class, including important analogues of constructions commonly found in operator algebras. In particular, for any (s-)unital $K$-algebra having enough nonzero idempotents (for example, for a von Neumann regular algebra) its tensor product over $K$ with many nonsimple Leavitt path algebras is purely infinite. },
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In this paper we introduce the concept of purely infinite rings, which in the simple case agrees with the already existing notion of pure infiniteness. We establish various permanence properties of this notion, with respect to passage to matrix rings, corners, and behaviour under extensions, so being purely infinite is preserved under Morita equivalence. We show that a wealth of examples falls into this class, including important analogues of constructions commonly found in operator algebras. In particular, for any (s-)unital $K$-algebra having enough nonzero idempotents (for example, for a von Neumann regular algebra) its tensor product over $K$ with many nonsimple Leavitt path algebras is purely infinite.
Abrams, Gene; Aranda-Pino, Gonzalo; Perera, Francesc; Siles-Molina, Mercedes
Chain conditions for Leavitt path algebras Journal Article
In: Forum Mathematicum, vol. 22, no. 1, pp. 95-114, 2010.
@article{AbrAraPerSil,
title = {Chain conditions for Leavitt path algebras},
author = {Gene Abrams and Gonzalo Aranda-Pino and Francesc Perera and Mercedes Siles-Molina},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/ChainConditions.pdf},
doi = {10.1515/FORUM.2010.005},
year = {2010},
date = {2010-10-01},
journal = {Forum Mathematicum},
volume = {22},
number = {1},
pages = {95-114},
abstract = {In this paper we give necessary and sufficient conditions on a row-finite graph $E$ so that the corresponding (not necessarily unital) Leavitt path K-algebra $L_K(E)$ is either artinian or noetherian from both a local and a categorical perspective. These extend the known results in the unital case to a much wider context. Besides the graph theoretic conditions, we provide in both situations isomorphisms between these algebras and appropriate direct sums of matrix rings over $K$ or $K[x,x^−1]$.},
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In this paper we give necessary and sufficient conditions on a row-finite graph $E$ so that the corresponding (not necessarily unital) Leavitt path K-algebra $L_K(E)$ is either artinian or noetherian from both a local and a categorical perspective. These extend the known results in the unital case to a much wider context. Besides the graph theoretic conditions, we provide in both situations isomorphisms between these algebras and appropriate direct sums of matrix rings over $K$ or $K[x,x^−1]$.
Kucerovsky, Dan; Ng, Ping Wong; Perera, Francesc
Purely infinite corona algebras of simple C*-algebras Journal Article
In: Mathematische Annalen, vol. 346, no. 1, pp. 23-40, 2010.
@article{KucNgPer,
title = {Purely infinite corona algebras of simple C*-algebras},
author = {Dan Kucerovsky and Ping Wong Ng and Francesc Perera},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/knpfinal2.pdf},
doi = {10.1007/s00208-009-0382-0},
year = {2010},
date = {2010-01-21},
journal = {Mathematische Annalen},
volume = {346},
number = {1},
pages = {23-40},
abstract = {In this paper, we study the problem of when the corona algebra of a non-unital C*-algebra is purely infinite. A complete answer is obtained for stabilisations of simple and unital algebras that have enough comparison of positive elements. Our result relates the pure infiniteness condition (from its strongest to weakest forms) to the geometry of the tracial simplex of the algebra, and to the behaviour of corona projections, despite the fact that there is no real rank zero condition.},
keywords = {},
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In this paper, we study the problem of when the corona algebra of a non-unital C*-algebra is purely infinite. A complete answer is obtained for stabilisations of simple and unital algebras that have enough comparison of positive elements. Our result relates the pure infiniteness condition (from its strongest to weakest forms) to the geometry of the tracial simplex of the algebra, and to the behaviour of corona projections, despite the fact that there is no real rank zero condition.
2009
Brešar, Matej; Perera, Francesc; Sánchez-Ortega, Juana; Siles-Molina, Mercedes
Computing the maximal algebra of quotients of a Lie algebra Journal Article
In: Forum Mathematicum, vol. 21, no. 4, pp. 601-620, 2009.
@article{BrePerSanSil,
title = {Computing the maximal algebra of quotients of a Lie algebra},
author = {Matej Brešar and Francesc Perera and Juana Sánchez-Ortega and Mercedes Siles-Molina},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/maximal_algebra_of_quotients.pdf},
doi = {10.1515/FORUM.2009.030},
year = {2009},
date = {2009-10-14},
journal = {Forum Mathematicum},
volume = {21},
number = {4},
pages = {601-620},
abstract = {The maximal algebra of quotients of a semiprime Lie algebra was introduced recently by M. Siles Molina. In the present paper we answer some natural questions concerning this concept, and describe maximal algebras of quotients of certain Lie algebras that arise from associative algebras.},
keywords = {},
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The maximal algebra of quotients of a semiprime Lie algebra was introduced recently by M. Siles Molina. In the present paper we answer some natural questions concerning this concept, and describe maximal algebras of quotients of certain Lie algebras that arise from associative algebras.
2008
Ara, Pere; Perera, Francesc; Wehrung, Friedrich
Finitely generated antisymmetric graph monoids Journal Article
In: Journal of Algebra, vol. 320, no. 5, pp. 1963-1982, 2008.
@article{AraPerWeh,
title = {Finitely generated antisymmetric graph monoids},
author = {Pere Ara and Francesc Perera and Friedrich Wehrung},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/GraphMon.pdf},
doi = {10.1016/j.jalgebra.2008.06.013},
year = {2008},
date = {2008-09-10},
journal = {Journal of Algebra},
volume = {320},
number = {5},
pages = {1963-1982},
abstract = {A graph monoid is a commutative monoid for which there is a particularly simple presentation, given in terms of a quiver. Such monoids are known to satisfy various nonstable K-theoretical representability properties for either von Neumann regular rings or C*-algebras. We give a characterization of graph monoids within finitely generated antisymmetric refinement monoids. This characterization is formulated in terms of the prime elements of the monoid, and it says that each free prime has at most one free lower cover. We also characterize antisymmetric graph monoids of finite quivers. In particular, the monoid $mathbb{Z}^infty={0,1,2,dots}cup{infty}$ is a graph monoid, but it is not the graph monoid of any finite quiver.},
keywords = {},
pubstate = {published},
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}
A graph monoid is a commutative monoid for which there is a particularly simple presentation, given in terms of a quiver. Such monoids are known to satisfy various nonstable K-theoretical representability properties for either von Neumann regular rings or C*-algebras. We give a characterization of graph monoids within finitely generated antisymmetric refinement monoids. This characterization is formulated in terms of the prime elements of the monoid, and it says that each free prime has at most one free lower cover. We also characterize antisymmetric graph monoids of finite quivers. In particular, the monoid $mathbb{Z}^infty={0,1,2,dots}cup{infty}$ is a graph monoid, but it is not the graph monoid of any finite quiver.
Brown, Nate P.; Perera, Francesc; Toms, Andrew S.
The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras Journal Article
In: Journal für die Reine und Angewandte Mathematik, vol. 621, pp. 191-211, 2008.
@article{BroPerTom,
title = {The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras},
author = {Nate P. Brown and Francesc Perera and Andrew S. Toms},
doi = {10.1515/CRELLE.2008.062},
year = {2008},
date = {2008-06-12},
journal = {Journal für die Reine und Angewandte Mathematik},
volume = {621},
pages = {191-211},
abstract = {We prove that the Cuntz semigroup is recovered functorially from the Elliott invariant for a large class of C*-algebras. In particular, our results apply to the largest class of simple C*-algebras for which K-theoretic classification can be hoped for. This work has three significant consequences. First, it provides new conceptual insight into Elliott's classification program, proving that the usual form of the Elliott conjecture is equivalent, among Z-stable algebras, to a conjecture which is in general substantially weaker and for which there are no known counterexamples. Second and third, it resolves, for the class of algebras above, two conjectures of Blackadar and Handelman concerning the basic structure of dimension functions on C*-algebras. We also prove in passing that the Cuntz-Pedersen semigroup is recovered functorially from the Elliott invariant for a large class of unital simple C*-algebras. },
keywords = {},
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We prove that the Cuntz semigroup is recovered functorially from the Elliott invariant for a large class of C*-algebras. In particular, our results apply to the largest class of simple C*-algebras for which K-theoretic classification can be hoped for. This work has three significant consequences. First, it provides new conceptual insight into Elliott's classification program, proving that the usual form of the Elliott conjecture is equivalent, among Z-stable algebras, to a conjecture which is in general substantially weaker and for which there are no known counterexamples. Second and third, it resolves, for the class of algebras above, two conjectures of Blackadar and Handelman concerning the basic structure of dimension functions on C*-algebras. We also prove in passing that the Cuntz-Pedersen semigroup is recovered functorially from the Elliott invariant for a large class of unital simple C*-algebras.
Perera, Francesc; Siles-Molina, Mercedes
Strongly non-degenerate Lie algebras Journal Article
In: Proceedings of the American Mathematical Society, vol. 136, no. 12, pp. 4115-4124, 2008.
@article{PerSil,
title = {Strongly non-degenerate Lie algebras},
author = {Francesc Perera and Mercedes Siles-Molina},
doi = {10.1090/S0002-9939-08-09558-0},
year = {2008},
date = {2008-05-01},
journal = {Proceedings of the American Mathematical Society},
volume = {136},
number = {12},
pages = {4115-4124},
abstract = {Let $A$ be a semiprime 2 and 3-torsion free non-commutative associative algebra. We show that the Lie algebra $der(A)$ of (associative) derivations of $A$ is strongly non-degenerate, which is a strong form of semiprimeness for Lie algebras, under some additional restrictions on the center of $A$. This result follows from a description of the quadratic annihilator of a general Lie algebra inside appropriate Lie overalgebras. Similar results are obtained for an associative algebra $A$ with involution and the Lie algebra $sder(A)$ of involution preserving derivations of $A$. },
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}
Let $A$ be a semiprime 2 and 3-torsion free non-commutative associative algebra. We show that the Lie algebra $der(A)$ of (associative) derivations of $A$ is strongly non-degenerate, which is a strong form of semiprimeness for Lie algebras, under some additional restrictions on the center of $A$. This result follows from a description of the quadratic annihilator of a general Lie algebra inside appropriate Lie overalgebras. Similar results are obtained for an associative algebra $A$ with involution and the Lie algebra $sder(A)$ of involution preserving derivations of $A$.
Perera, Francesc; Siles-Molina, Mercedes
Associative and Lie algebras of quotients Journal Article
In: Publicacions Matemàtiques, vol. 52, no. 1, pp. 129-149, 2008.
@article{PerSil:Lie,
title = {Associative and Lie algebras of quotients },
author = {Francesc Perera and Mercedes Siles-Molina},
doi = {10.5565/PUBLMAT_52108_06},
year = {2008},
date = {2008-01-16},
journal = {Publicacions Matemàtiques},
volume = {52},
number = {1},
pages = {129-149},
abstract = {In this paper we examine how the notion of algebra of quotients for Lie algebras ties up with the corresponding well-known concept in the associative case. Specifically, we completely characterize when a Lie algebra $Q$ is an algebra of quotients of a Lie algebra $L$ in terms of the associative algebras generated by the adjoint operators of $L$ and $Q$ respectively. In a converse direction, we also provide with new examples of algebras of quotients of Lie algebras and these come from associative algebras of quotients. In the course of our analysis, we make use of the notions of density and multiplicative semiprimeness to link our results with the maximal symmetric ring of quotients. },
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper we examine how the notion of algebra of quotients for Lie algebras ties up with the corresponding well-known concept in the associative case. Specifically, we completely characterize when a Lie algebra $Q$ is an algebra of quotients of a Lie algebra $L$ in terms of the associative algebras generated by the adjoint operators of $L$ and $Q$ respectively. In a converse direction, we also provide with new examples of algebras of quotients of Lie algebras and these come from associative algebras of quotients. In the course of our analysis, we make use of the notions of density and multiplicative semiprimeness to link our results with the maximal symmetric ring of quotients.
2007
Ara, Pere; Perera, Francesc
Non-stable K-theory for QB-rings Journal Article
In: Mathematica Scandinavica, vol. 100, no. 2, pp. 265-300, 2007.
@article{AraPer:Nonstable,
title = {Non-stable K-theory for QB-rings},
author = {Pere Ara and Francesc Perera},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/nonstable.pdf},
doi = {10.7146/math.scand.a-15024},
year = {2007},
date = {2007-10-03},
journal = {Mathematica Scandinavica},
volume = {100},
number = {2},
pages = {265-300},
abstract = {We study the class of QB-rings that satisfy the weak cancellation condition of separativity for finitely generated projective modules. This property turns out to be crucial for proving that all (quasi-)invertible matrices over a QB-ring can be diagonalised using row and column operations. The main two consequences of this fact are: (i) The natural map GL_1(R)→K_1(R) is surjective, and (ii) the only obstruction to lift invertible elements from a quotient is of K-theoretical nature. We also show that for a reasonably large class of QB-rings that includes the prime ones, separativity always holds.},
keywords = {},
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tppubtype = {article}
}
We study the class of QB-rings that satisfy the weak cancellation condition of separativity for finitely generated projective modules. This property turns out to be crucial for proving that all (quasi-)invertible matrices over a QB-ring can be diagonalised using row and column operations. The main two consequences of this fact are: (i) The natural map GL_1(R)→K_1(R) is surjective, and (ii) the only obstruction to lift invertible elements from a quotient is of K-theoretical nature. We also show that for a reasonably large class of QB-rings that includes the prime ones, separativity always holds.
Perera, Francesc; Toms, Andrew S.
Recasting the Elliott conjecture Journal Article
In: Mathematische Annalen, vol. 338, no. 3, pp. 669-702, 2007.
@article{PerTom,
title = {Recasting the Elliott conjecture},
author = {Francesc Perera and Andrew S. Toms},
doi = {10.1007/s00208-007-0093-3},
year = {2007},
date = {2007-05-16},
journal = {Mathematische Annalen},
volume = {338},
number = {3},
pages = {669-702},
abstract = {Let A be a simple, unital, exact, and finite C*-algebra which absorbs the Jiang-Su algebra Z tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases -- Z-stable algebras all -- we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of Z-stable algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott's classification conjecture -- that K-theoretic invariants will classify separable and nuclear C*-algebras -- with the recent appearance of counterexamples to its strongest concrete form. },
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Let A be a simple, unital, exact, and finite C*-algebra which absorbs the Jiang-Su algebra Z tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases -- Z-stable algebras all -- we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of Z-stable algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott's classification conjecture -- that K-theoretic invariants will classify separable and nuclear C*-algebras -- with the recent appearance of counterexamples to its strongest concrete form.
2005
Pedersen, Gert K.; Perera, Francesc
Inverse limits of rings and multiplier rings Journal Article
In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 139, no. 2, pp. 207-228, 2005.
@article{PedPer,
title = {Inverse limits of rings and multiplier rings},
author = {Gert K. Pedersen and Francesc Perera},
doi = {10.1017/S0305004105008704},
year = {2005},
date = {2005-10-12},
journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
volume = {139},
number = {2},
pages = {207-228},
abstract = {It is proved that the exchange property, the Bass stable rank and the quasi-Bass property are all preserved under surjective inverse limits. This is then applied to multiplier rings by showing that in many cases can be obtained as inverse limits.
},
keywords = {},
pubstate = {published},
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It is proved that the exchange property, the Bass stable rank and the quasi-Bass property are all preserved under surjective inverse limits. This is then applied to multiplier rings by showing that in many cases can be obtained as inverse limits.
Ara, Pere; Pedersen, Gert K.; Perera, Francesc
Extensions and pullbacks in QB-rings Journal Article
In: Algebras and Representation Theory, vol. 8, no. 1, pp. 75-97, 2005.
@article{AraPedPer,
title = {Extensions and pullbacks in QB-rings},
author = {Pere Ara and Gert K. Pedersen and Francesc Perera},
doi = {10.1007/s10468-004-5767-x},
year = {2005},
date = {2005-06-16},
journal = {Algebras and Representation Theory},
volume = {8},
number = {1},
pages = {75-97},
abstract = {We prove a new extension result for $QB-$rings that allows us to examine extensions of rings where the ideal is purely infinite and simple. We then use this result to explore various constructions that provide new examples of $QB-$rings. More concretely, we show that a surjective pullback of two $QB-$rings is usually again a $QB-$ring. Specializing to the case of an extension of a semi-prime ideal $I$ of a unital ring $R$, the pullback setting leads naturally to the study of rings whose multiplier rings are $QB-$rings. For a wide class of regular rings, we give necessary and sufficient conditions for their multiplier rings to be $QB-$rings. Our analysis is based on the study of extensions and the use of non-stable $K-$theoretical techniques. },
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We prove a new extension result for $QB-$rings that allows us to examine extensions of rings where the ideal is purely infinite and simple. We then use this result to explore various constructions that provide new examples of $QB-$rings. More concretely, we show that a surjective pullback of two $QB-$rings is usually again a $QB-$ring. Specializing to the case of an extension of a semi-prime ideal $I$ of a unital ring $R$, the pullback setting leads naturally to the study of rings whose multiplier rings are $QB-$rings. For a wide class of regular rings, we give necessary and sufficient conditions for their multiplier rings to be $QB-$rings. Our analysis is based on the study of extensions and the use of non-stable $K-$theoretical techniques.
Ortus, Francisco; Pardo, Enric; Perera, Francesc
Simple Riesz groups of rank one having wild intervals Journal Article
In: Journal of Algebra, vol. 284, no. 1, pp. 111-140, 2005.
BibTeX | Links:
@article{OrtParPer,
title = {Simple Riesz groups of rank one having wild intervals},
author = {Francisco Ortus and Enric Pardo and Francesc Perera},
doi = {10.1016/j.jalgebra.2004.10.002 },
year = {2005},
date = {2005-02-15},
journal = {Journal of Algebra},
volume = {284},
number = {1},
pages = {111-140},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
2004
Perera, Francesc; Rørdam, Mikael
AF-embeddings into C*-algebras of real rank zero Journal Article
In: Journal of Functional Analysis, vol. 217, no. 1, pp. 142-170, 2004.
@article{PerRor,
title = {AF-embeddings into C*-algebras of real rank zero},
author = {Francesc Perera and Mikael Rørdam},
doi = {10.1016/j.jfa.2004.05.001},
year = {2004},
date = {2004-10-22},
journal = {Journal of Functional Analysis},
volume = {217},
number = {1},
pages = {142-170},
abstract = {It is proved that every separable C*-algebra of real rank zero contains an AF-sub-C*-algebra such that the inclusion mapping induces an isomorphism of the ideal lattices of the two C*-algebras and such that every projection in a matrix algebra over the large C*-algebra is equivalent to a projection in a matrix algebra over the AF-sub-C*-algebra. This result is proved at the level of monoids, using that the monoid of Murray-von Neumann equivalence classes of projections in a C*-algebra of real rank zero has the refinement property. As an application of our result, we show that given a unital C*-algebra $A$ of real rank zero and a natural number $n$, then there is a unital *-homomorphism $M_{n_1} oplus ... oplus M_{n_r} to A$ for some natural numbers $r,n_1, ...,n_r$ with $n_j ge n$ for all $j$ if and only if $A$ has no representation of dimension less than $n$. },
keywords = {},
pubstate = {published},
tppubtype = {article}
}
It is proved that every separable C*-algebra of real rank zero contains an AF-sub-C*-algebra such that the inclusion mapping induces an isomorphism of the ideal lattices of the two C*-algebras and such that every projection in a matrix algebra over the large C*-algebra is equivalent to a projection in a matrix algebra over the AF-sub-C*-algebra. This result is proved at the level of monoids, using that the monoid of Murray-von Neumann equivalence classes of projections in a C*-algebra of real rank zero has the refinement property. As an application of our result, we show that given a unital C*-algebra $A$ of real rank zero and a natural number $n$, then there is a unital *-homomorphism $M_{n_1} oplus ... oplus M_{n_r} to A$ for some natural numbers $r,n_1, ...,n_r$ with $n_j ge n$ for all $j$ if and only if $A$ has no representation of dimension less than $n$.
Ara, Pere; O’Meara, Kevin C.; Perera, Francesc
Gromov translation algebras over discrete trees are exchange rings Journal Article
In: Transactions of the American Mathematical Society, vol. 356, no. 5, pp. 2067-2079, 2004.
@article{AraOMePer,
title = {Gromov translation algebras over discrete trees are exchange rings},
author = {Pere Ara and Kevin C. O’Meara and Francesc Perera},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/translation_algebras.pdf},
doi = {10.1090/S0002-9947-03-03372-5},
year = {2004},
date = {2004-05-20},
journal = {Transactions of the American Mathematical Society},
volume = {356},
number = {5},
pages = {2067-2079},
abstract = {It is shown that the Gromov translation ring of a discrete tree over a von Neumann regular ring is an exchange ring. This provides a new source of exchangerings, including for example, the algebras G(0) of ω×ω matrices (over a field) of constant bandwidth. An extension of these ideas shows that for all real numbers r in the unit interval [0,1], the growth algebras G(r) (introduced by Hannah and O’Meara in 1993) are exchange rings. Consequently, over a countable field, countable-dimensional exchange algebras can take any prescribed bandwidth dimension r in [0,1].},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
It is shown that the Gromov translation ring of a discrete tree over a von Neumann regular ring is an exchange ring. This provides a new source of exchangerings, including for example, the algebras G(0) of ω×ω matrices (over a field) of constant bandwidth. An extension of these ideas shows that for all real numbers r in the unit interval [0,1], the growth algebras G(r) (introduced by Hannah and O’Meara in 1993) are exchange rings. Consequently, over a countable field, countable-dimensional exchange algebras can take any prescribed bandwidth dimension r in [0,1].
2002
Ara, Pere; O’Meara, Kevin C.; Perera, Francesc
Stable finiteness of group rings in arbitrary characteristic Journal Article
In: Advances in Mathematics, vol. 170, no. 2, pp. 224-238, 2002.
@article{AraOMePer:Stable,
title = {Stable finiteness of group rings in arbitrary characteristic},
author = {Pere Ara and Kevin C. O’Meara and Francesc Perera},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/stable_finiteness.pdf},
doi = {10.1006/aima.2002.2075},
year = {2002},
date = {2002-05-23},
journal = {Advances in Mathematics},
volume = {170},
number = {2},
pages = {224-238},
abstract = {We show that every (discrete) group ring D[G] of a free-by-amenable group G over a division ring D of arbitrary characteristic is stably finite, in the sense that one-sided inverses in all matrix rings over D[G] are two-sided. Our methods use Sylvester rank functions and the translation ring of an amenable group.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We show that every (discrete) group ring D[G] of a free-by-amenable group G over a division ring D of arbitrary characteristic is stably finite, in the sense that one-sided inverses in all matrix rings over D[G] are two-sided. Our methods use Sylvester rank functions and the translation ring of an amenable group.
2001
Ara, Pere; Pedersen, Gert K.; Perera, Francesc
A closure operation in rings Journal Article
In: International Journal of Mathematics, vol. 12, no. 7, pp. 791-812, 2001.
@article{AraPerPed:closure,
title = {A closure operation in rings},
author = {Pere Ara and Gert K. Pedersen and Francesc Perera},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/closure.pdf},
doi = {10.1142/S0129167X01001039},
year = {2001},
date = {2001-10-04},
journal = {International Journal of Mathematics},
volume = {12},
number = {7},
pages = {791-812},
abstract = {We study the operation E→cl(E) defined on subsets E of a unital ring R, where x∈cl(E) if (x+Rb)∩E≠∅ for each b in R such that Rx+Rb=R. This operation, which strongly resembles a closure, originates in algebraic K-theory. For any left ideal L we show that cl(L) equals the intersection of the maximal left ideals of R containing L. Moreover, cl(Re)=Re+rad(R) if e is an idempotent in R, and cl(I)=I for a two-sided ideal I precisely when I is semi-primitive in R (i.e. rad(R/I)=0). We then explore a special class of von Neumann regular elements in R, called persistently regular and characterized by forming an “open” subset R pr in R, i.e. cl(R∖R pr )=R∖R pr . In fact, R∖R pr =cl(R∖R r ), so that R pr is the “algebraic interior” of the set R r of regular elements. We show that a regular element x with partial inverse y is persistently regular, if and only if the skew corner (1-xy)R(1-yx) is contained in R r . If I reg (R) denotes the maximal regular ideal in R and R q -1 the set of quasi-invertible elements, defined and studied in [the authors, J. Algebra 230, No. 2, 608-655 (2000; Zbl 0963.16008)], we prove that R q -1 +I reg (R)⊂R pr . Specializing to C * -algebras we prove that cl(E) coincides with the norm closure of E, when E is one of the five interesting sets R^-1 , R_ℓ^-1 , R_ r^-1 , R_q^-1 and R_sa^-1 , and that R_pr coincides with the topological interior of R_r . We also show that the operation cl respects boundedness, self-adjointness and positivity.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study the operation E→cl(E) defined on subsets E of a unital ring R, where x∈cl(E) if (x+Rb)∩E≠∅ for each b in R such that Rx+Rb=R. This operation, which strongly resembles a closure, originates in algebraic K-theory. For any left ideal L we show that cl(L) equals the intersection of the maximal left ideals of R containing L. Moreover, cl(Re)=Re+rad(R) if e is an idempotent in R, and cl(I)=I for a two-sided ideal I precisely when I is semi-primitive in R (i.e. rad(R/I)=0). We then explore a special class of von Neumann regular elements in R, called persistently regular and characterized by forming an “open” subset R pr in R, i.e. cl(R∖R pr )=R∖R pr . In fact, R∖R pr =cl(R∖R r ), so that R pr is the “algebraic interior” of the set R r of regular elements. We show that a regular element x with partial inverse y is persistently regular, if and only if the skew corner (1-xy)R(1-yx) is contained in R r . If I reg (R) denotes the maximal regular ideal in R and R q -1 the set of quasi-invertible elements, defined and studied in [the authors, J. Algebra 230, No. 2, 608-655 (2000; Zbl 0963.16008)], we prove that R q -1 +I reg (R)⊂R pr . Specializing to C * -algebras we prove that cl(E) coincides with the norm closure of E, when E is one of the five interesting sets R^-1 , R_ℓ^-1 , R_ r^-1 , R_q^-1 and R_sa^-1 , and that R_pr coincides with the topological interior of R_r . We also show that the operation cl respects boundedness, self-adjointness and positivity.
Perera, Francesc
Ideal structure of multiplier algebras of simple C*-algebras with real rank zero Journal Article
In: Canadian Journal of Mathematics, vol. 53, no. 3, pp. 592-630, 2001.
@article{Per:Canadian,
title = {Ideal structure of multiplier algebras of simple C*-algebras with real rank zero},
author = {Francesc Perera},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/multiplier_algebras.pdf},
doi = {10.4153/CJM-2001-025-2},
year = {2001},
date = {2001-05-17},
journal = {Canadian Journal of Mathematics},
volume = {53},
number = {3},
pages = {592-630},
abstract = {We give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of σ-unital simple C*-algebras A with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of the multiplier algebra M(A), is therefore analyzed. In important cases it is shown that, if A has finite scale then the quotient of M(A) modulo any closed ideal I that properly contains A has stable rank one. The intricacy of the ideal structure of M(A) is reflected in the fact that M(A) can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.},
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We give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of σ-unital simple C*-algebras A with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of the multiplier algebra M(A), is therefore analyzed. In important cases it is shown that, if A has finite scale then the quotient of M(A) modulo any closed ideal I that properly contains A has stable rank one. The intricacy of the ideal structure of M(A) is reflected in the fact that M(A) can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.
2000
Perera, Francesc
Algebraic aspects in the classification of C*-algebras Journal Article
In: Irish Mathematical Society Bulletin, vol. 45, pp. 33-55, 2000.
@article{Per:Irish,
title = {Algebraic aspects in the classification of C*-algebras},
author = {Francesc Perera},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/IrishBull.pdf},
year = {2000},
date = {2000-10-26},
journal = {Irish Mathematical Society Bulletin},
volume = {45},
pages = {33-55},
abstract = {We survey some recent results concerning the use of non-stable K-theoretic methods to efficiently analyse the ideal structure of multiplier algebras for a wide class of C*-algebras having real rank zero and stable rank one. Some applications of these results are delineated, showing a high degree of infiniteness of these objects.},
keywords = {},
pubstate = {published},
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}
We survey some recent results concerning the use of non-stable K-theoretic methods to efficiently analyse the ideal structure of multiplier algebras for a wide class of C*-algebras having real rank zero and stable rank one. Some applications of these results are delineated, showing a high degree of infiniteness of these objects.
Perera, Francesc
Extremal richness of multiplier and corona algebras of simple C*-algebras with real rank zero Journal Article
In: Journal of Operator Theory, vol. 44, no. 2, pp. 413-431, 2000.
@article{Per:extremal,
title = {Extremal richness of multiplier and corona algebras of simple C*-algebras with real rank zero},
author = {Francesc Perera},
year = {2000},
date = {2000-10-04},
journal = {Journal of Operator Theory},
volume = {44},
number = {2},
pages = {413-431},
abstract = {In this paper we investigate the extremal richness of the multiplier algebra $M(A)$ and the corona algebra $M(A)/A$, for a simple C*-algebra $A$ with real rank zero and stable rank one. We show that the space of extremal quasitraces and the scale of $A$ contain enough information to determine whether $M(A)/A$ is extremally rich. In detail, if the scale is finite, then $M(A)/A$ is extremally rich. In important cases, and if the scale is not finite, extremal richness is characterized by a restrictive condition: the existence of only one infinite extremal quasitrace which is isolated in a convex sense. },
keywords = {},
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}
In this paper we investigate the extremal richness of the multiplier algebra $M(A)$ and the corona algebra $M(A)/A$, for a simple C*-algebra $A$ with real rank zero and stable rank one. We show that the space of extremal quasitraces and the scale of $A$ contain enough information to determine whether $M(A)/A$ is extremally rich. In detail, if the scale is finite, then $M(A)/A$ is extremally rich. In important cases, and if the scale is not finite, extremal richness is characterized by a restrictive condition: the existence of only one infinite extremal quasitrace which is isolated in a convex sense.
Ara, Pere; Perera, Francesc
Multipliers of von Neumann regular rings Journal Article
In: Communications in Algebra, vol. 28, no. 7, pp. 3359-3385, 2000.
@article{AraPer:multipliers,
title = {Multipliers of von Neumann regular rings},
author = {Pere Ara and Francesc Perera},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/multiplier_rings.pdf},
doi = {10.1080/00927870008827031},
year = {2000},
date = {2000-09-28},
journal = {Communications in Algebra},
volume = {28},
number = {7},
pages = {3359-3385},
abstract = {We analyse the structure of the multiplier ring M(R) of a (nonunital) Von Neumann regular ring R. We show that M(R) is not regular in general, but every principal right ideal is generated by two idempotents. This, together with Riesz Decomposition on idempotents of M(R), furnishes a description of the monoid V(M(R)) of Murray-Von Neumann equivalence classes of idempotents which is used to examine efficiently the lattice of ideals of M(R). The techniques developed here will allow as well other applications to the category of projective modules over regular rings.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We analyse the structure of the multiplier ring M(R) of a (nonunital) Von Neumann regular ring R. We show that M(R) is not regular in general, but every principal right ideal is generated by two idempotents. This, together with Riesz Decomposition on idempotents of M(R), furnishes a description of the monoid V(M(R)) of Murray-Von Neumann equivalence classes of idempotents which is used to examine efficiently the lattice of ideals of M(R). The techniques developed here will allow as well other applications to the category of projective modules over regular rings.
Ara, Pere; Pedersen, Gert K.; Perera, Francesc
An infinite analogue of rings with stable rank one Journal Article
In: Journal of Algebra, vol. 230, no. 2, pp. 608-655, 2000.
BibTeX | Links:
@article{AraPedPer:QB,
title = {An infinite analogue of rings with stable rank one},
author = {Pere Ara and Gert K. Pedersen and Francesc Perera},
doi = {10.1006/jabr.2000.8330},
year = {2000},
date = {2000-08-22},
journal = {Journal of Algebra},
volume = {230},
number = {2},
pages = {608-655},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Perera, Francesc
Lifting units modulo exchange ideals and C*-algebras with real rank zero Journal Article
In: Journal für die Reine und Angewandte Mathematik, vol. 522, pp. 51-62, 2000.
@article{Per:Crelle,
title = {Lifting units modulo exchange ideals and C*-algebras with real rank zero},
author = {Francesc Perera},
doi = {10.1515/crll.2000.040},
year = {2000},
date = {2000-07-20},
journal = {Journal für die Reine und Angewandte Mathematik},
volume = {522},
pages = {51-62},
abstract = {Given a unital ring $R$ and a two-sided ideal $I$ of $R$, we consider the question of determining when a unit of $R/I$ can be lifted to a unit of $R$. For the wide class of separative exchange ideals $I$, we show that the only obstruction to lifting invertibles relies on a K-theoretic condition on $I$. This allows to extend previously known index theories to this context. Using this we can draw consequences for von Neumann regular rings and C*-algebras with real rank zero. },
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Given a unital ring $R$ and a two-sided ideal $I$ of $R$, we consider the question of determining when a unit of $R/I$ can be lifted to a unit of $R$. For the wide class of separative exchange ideals $I$, we show that the only obstruction to lifting invertibles relies on a K-theoretic condition on $I$. This allows to extend previously known index theories to this context. Using this we can draw consequences for von Neumann regular rings and C*-algebras with real rank zero.
Ara, Pere; Pardo, Enric; Perera, Francesc
The structure of countably generated projective modules over regular rings Journal Article
In: Journal of Algebra, vol. 226, no. 1, pp. 161-190, 2000.
@article{AraParPer,
title = {The structure of countably generated projective modules over regular rings},
author = {Pere Ara and Enric Pardo and Francesc Perera},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/Countgen.pdf},
doi = {10.1006/jabr.1999.8156},
year = {2000},
date = {2000-06-13},
journal = {Journal of Algebra},
volume = {226},
number = {1},
pages = {161-190},
abstract = {We prove that, for every regular ring R, there exists an isomorphism between the monoids of isomorphism classes of finitely generated projective right modules over the rings End_R(R^(ω)_R) and RCFM(R), where the latter denotes the ring of countably infinite row- and column-finite matrices over R. We use this result to give a precise description of the countably generated projective modules over simple regular rings and over regular rings satisfying s-comparability.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We prove that, for every regular ring R, there exists an isomorphism between the monoids of isomorphism classes of finitely generated projective right modules over the rings End_R(R^(ω)_R) and RCFM(R), where the latter denotes the ring of countably infinite row- and column-finite matrices over R. We use this result to give a precise description of the countably generated projective modules over simple regular rings and over regular rings satisfying s-comparability.
1999
Perera, Francesc
Monoids arising from positive matrices over commutative C*-algebras Journal Article
In: Mathematical Proceedings of the Royal Irish Academy, vol. 99A, no. 1, pp. 75-84, 1999.
@article{Per:MonMatrix,
title = {Monoids arising from positive matrices over commutative C*-algebras},
author = {Francesc Perera},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/commutative.pdf},
year = {1999},
date = {1999-10-13},
journal = {Mathematical Proceedings of the Royal Irish Academy},
volume = {99A},
number = {1},
pages = {75-84},
abstract = {We study the structure of the Cuntz monoid S(C_0(X)) associated with a locally compact space X. It is shown that for a wide class of spaces, namely locally compact σ-compact spaces, the Riesz decomposition on the monoid forces the (covering) dimension of the space to be zero. It is possible then to diagonalise matrices over C_0(X) in a unique way with respect to an equivalence relation. We give a representation of S(C_0(X)) as a monoid of lower semicontinuous functions over X, from which order-cancellation on S(C_0(X)) follows.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study the structure of the Cuntz monoid S(C_0(X)) associated with a locally compact space X. It is shown that for a wide class of spaces, namely locally compact σ-compact spaces, the Riesz decomposition on the monoid forces the (covering) dimension of the space to be zero. It is possible then to diagonalise matrices over C_0(X) in a unique way with respect to an equivalence relation. We give a representation of S(C_0(X)) as a monoid of lower semicontinuous functions over X, from which order-cancellation on S(C_0(X)) follows.
1997
Perera, Francesc
The structure of positive elements for C*-algebras with real rank zero Journal Article
In: International Journal of Mathematics, vol. 8, no. 3, pp. 383-405, 1997.
@article{Per:RR0,
title = {The structure of positive elements for C*-algebras with real rank zero},
author = {Francesc Perera},
url = {http://mat.uab.cat/web/perera/wp-content/uploads/sites/16/2019/10/positive.pdf},
doi = {10.1142/S0129167X97000196},
year = {1997},
date = {1997-10-23},
journal = {International Journal of Mathematics},
volume = {8},
number = {3},
pages = {383-405},
abstract = {In this paper we give a representation theorem for the Cuntz monoid S(A) of a σ-unital C*-algebra A with real rank zero and stable rank one, which allows to prove several Riesz decomposition properties on the monoid. As a consequence, it is proved that the comparability conditions (FCQ), stable (FCQ) and (FCQ+) are equivalent for simple C*-algebras with real rank zero. It is also shown that the Grothendieck group of S(A) is a Riesz group, and lattice-ordered under some additional assumptions on A.},
keywords = {},
pubstate = {published},
tppubtype = {article}
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In this paper we give a representation theorem for the Cuntz monoid S(A) of a σ-unital C*-algebra A with real rank zero and stable rank one, which allows to prove several Riesz decomposition properties on the monoid. As a consequence, it is proved that the comparability conditions (FCQ), stable (FCQ) and (FCQ+) are equivalent for simple C*-algebras with real rank zero. It is also shown that the Grothendieck group of S(A) is a Riesz group, and lattice-ordered under some additional assumptions on A.