Joan Bosa PhD in Mathematics

Publications

Preprints


P. Ara, R. Antoine, J.Bosa, F. Perera and E. Vilalta, "The Cuntz Semigroup of a Ring". Preprint 43pp. [pdf]

View Abstract>>

For any ring R, we introduce an invariant in the form of a partially ordered abelian semigroup S(R) built from an equivalence relation on the class of countably generated projective modules. We call S(R) the Cuntz semigroup of the ring R. This construction is akin to the manufacture of the Cuntz semigroup of a C*-algebra using countably generated Hilbert modules. To circumvent the lack of a topology in a general ring R, we deepen our understanding of countably projective modules over R, thus uncovering new features in their direct limit decompositions, which in turn yields two equivalent descriptions of S(R). The Cuntz semigroup of R is part of a new invariant SCu(R) which includes an ambient semigroup in the category of abstract Cuntz semigroups that provides additional information. We provide computations for both S(R) and SCu(R) in a number of interesting situations, such as unit-regular rings, semilocal rings, and in the context of nearly simple domains. We also relate our construcion to the Cuntz semigroup of a C*-algebra.

J.Bosa, A. Tikuisis and S. White, "Classifying Tracially factorizable maps from Nuclear cones". Preprint 12pp. [pdf]

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The main aim of these notes is to prove a uniqueness theorem for certain order zero maps into sequence algebras.

In press:


Published:


J.Bosa, "Stable Elements and Property (S)". J. of Mathematical Analysis and Applications, 512-2 (2022), 17pp. doi.org/10.1016/j.jmaa.2022.126171 [pdf]

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We study the relation (and differences) between stability and Property (S) in the simple and stably finite framework. This leads us to characterize stable elements in terms of its support, and study these concepts from different sides : hereditary subalgebras, projections in the multiplier algebra and order properties in the Cuntz semigroup. We use these approaches to show both that cancellation at infinity on the Cuntz semigroup just holds when its Cuntz equivalence is given by isomorphism at the level of Hilbert right-modules, and that different notions as Regularity, ω-comparison, Corona Factorization Property, property R, etc.. are equivalent under mild assumptions.

J.Bosa, J. Gabe, A. Sims and S. White, "The Nuclear Dimension of O_\infty-stable C*-algebras". Adv. in Mathematics, 401 (2022) 108250, 51pp. doi.org/10.1016/j.aim.2022.108250 [pdf]

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We show that every nuclear O_\infty -stable *-homomorphism with a separable exact domain has nuclear dimension at most 1. In particular separable, nuclear, O_\infty -stable C*-algebras have nuclear dimension 1. We also characterise when O_\infty -stable C* -algebras have finite decomposition rank in terms of quasidiagonality and primitive-ideal structure, and determine when full O 2 -stable *-homomorphisms have nuclear dimension 0.

P. Ara, C. Bonicke, J.Bosa and K. Li, "The Type Semigroup, Comparison and Almost finiteness for ample groupoids". Ergodic Theory and Dynamical Systems, (2021) 1-40. doi:10.1017/etds.2021.115 [pdf]

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In this article, we prove that a minimal ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions.

P. Ara, J.Bosa, E. Pardo and A. Sims, "The Groupoids of Adaptable Separated graphs and Their Type semigroups." Int. Mat. Res. Not. (IMRN), Vol 2021, No. 20, (2021), pp. 15444-15496. doi/10.1093/imrn/rnaa022 -- [pdf]

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Given an adaptable separated graph, we construct an associated groupoid and explore its type semigroup. Specifically, we first attach to each adaptable separated graph a corresponding semigroup, which we prove is an E* -unitary inverse semigroup. As a con- sequence, the tight groupoid of this semigroup is a Hausdorff étale groupoid. We show that this groupoid is always amenable, and that the type semigroups of groupoids obtained from adaptable separated graphs in this way include all finitely generated conical refinement monoids. The first three named authors will utilize this construction in forthcoming work to solve the Realization Problem for von Neumann regular rings, in the finitely generated case.

P. Ara, J.Bosa and E. Pardo, "The realization problem for finitely generated refinement monoids", Selecta Mathematica, New Series, 2020, 26(3), Paper No. 33, 63 pp [pdf]

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We show that every finitely generated conical refinement monoid can be rep- resented as the monoid V(R) of isomorphism classes of finitely generated projective mod- ules over a von Neumann regular ring R. To this end, we use the representation of these monoids provided by adaptable separated graphs. Given an adaptable separated graph (E, C) and a field K, we build a von Neumann regular K-algebra Q_K (E, C) and show that there is a natural isomorphism between the separated graph monoid M(E, C) and the monoid V(Q_K (E, C)).

P. Ara, J.Bosa and E. Pardo "Refinement monoids and adaptable separated graphs," Semigroup Forum, 2020, 101(1), pp. 19-36. [pdf]

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We define a subclass of separated graphs, the class of adaptable separated graphs, and study their associated monoids. We show that these monoids are primely generated conical refinement monoids, and we explicitly determine their associated I-systems. We also show that any finitely generated conical refinement monoid can be represented as the monoid of an adaptable separated graph. These results provide the first step toward an affirmative answer to the Realization Problem for von Neumann regular rings, in the finitely generated case.

P. Ara, C. Bonicke, J.Bosa and K. Li, "Strict Comparison for C*-algebras arising from Almost finite groupoids". Banach J. of Math. Analysis, 2020, 14(4), pp.1692-1710. [pdf]

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In this paper, we study Matui’s notion of almost finiteness for ample groupoids. If G is an almost finite minimal ample groupoid, we show that its reduced C∗-algebra C_r^∗ (G) has real rank zero and strict comparison even though C_r^∗ (G) may not be nuclear in general. If G is also second countable and non-elementary, then its Cuntz semigroup Cu(C_r^∗ (G)) is almost divisible and Cu(C^∗_r (G))\cong Cu(C^∗_r (G)\otimes Z) are canonically order-isomorphic, where Z denotes the Jiang-Su algebra.

J.Bosa, G. Tornetta and J. Zacharias, "A Bivariant Theory for the Cuntz Semigroup" Journal of Functional Analysis, 277 (2019), num 4, pp. 1061-111. [pdf]

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We introduce a bivariant version of the Cuntz semigroup as equivalence classes of order zero maps generalising the ordinary Cuntz semigroup. The theory has many properties formally analogous to $KK$-theory including a composition product. We establish basic properties concerning additivity, stability and continuity and study categorical properties in the setting of local C*-algebras. We determine the bivariant Cuntz semigroup for numerous examples such as when the second algebra is a Kirchberg algebra, and Cuntz homology for compact Hausdorff spaces which provides a complete invariant. Moreover we establish identities when tensoring with strongly self-absorbing C*-algebras. Finally we show that the bivariant Cuntz semigroup of the present work can be used to classify all unital and stably finite C*-algebras.

J.Bosa, N. Brown, Y. Sato, A. Tikuisis, S. White, W. Winter, "Covering Dimension of C*-algebras and 2-coloured classification", Memoirs of the American Mathematical Society, vol 257, num. 1233 (2019), 97pp. [pdf]

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We introduce the concept of finitely coloured equivalence for unital *-homomorphisms between C*-algebras, for which unitary equivalence is the 1-coloured case. We use this notion to classify *-homomorphisms from separable, unital, nuclear C*-algebras into ultrapowers of simple, unital, nuclear, Z-stable C*-algebras with compact extremal trace space up to 2-coloured equivalence by their behaviour on traces; this is based on a 1-coloured classication theorem for certain order zero maps, also in terms of tracial data.

As an application we calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, Z-stable C*-algebras with compact extremal trace space: it is 1.In the case that the extremal trace space also has finite topological covering dimension, this confirms the remaining open implication of the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry and topology, we derive a "homotopy equivalence implies isomorphism" result for large classes of C*-algebras with finite nuclear dimension.

J.Bosa and M.Christensen, "The failure of the Corona Factorization Property for the Villadsen Algebra V_\infty", Mathematica Scandinavica, 123-1, (2018), pp 142-146. [pdf]

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we prove that the Villadsen algebra V_\infty does not satisfy the Corona Factorization Property (CFP).

J.Bosa and H. Petzka, "Comparison properties of the Cuntz semigroup and applications to C*-algebras", Canadian J. of Mathematics, 70 (2018), no. 1, pp 26-52. [pdf]

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We study comparison properties in the category Cu aiming to lift results to the C*-algebraic setting. We introduce a new comparison property and relate it to both the CFP and w-comparison. We show differences of all properties by providing examples, which suggest that the corona factorization property for C*-algebras might allow for both finite and infinite projections. In addition, we show that Rørdam’s simple, nuclear C*-algebra with a finite and an infinite projection does not have the CFP

J.Bosa, G. Tornetta and J. Zacharias, "Open projections and suprema in the Cutnz semigroup", Mathematical Proceedings Cambridge Philosophical Society 164 (2018), no. 1, 135-146. [pdf]

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We provide a new and concise proof of the existence of suprema in the Cuntz semigroup using the open projection picture of the Cuntz semigroup initiated in [ORT-11]. Our argument is based on the observation that the supremum of a countable set of open projections in the bidual of a C*-algebra A is again open and corresponds to the generated hereditary sub-C*-algebra of A.

P. Ara, J.Bosa, R. Hazrat and A. Sims, "Reconstruction of graded groupoids from graded Steinberg algebras", Forum Mathematicum 29 (2017), no.5, pp. 1023-1037. [pdf]

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We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally-graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functiosn supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies C*-isomorphism of C*-algebras for graphs E and F in which every cycle has an exit.

J.Bosa and M. Dadarlat, "Local triviality for continuous field C*-algebras", International Mathematics Research Notices (IMRN) 4 (2015), pp 1044-1055. [pdf]

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Let $X$ be a finite dimensional compact metrizable space. Let $A$ be a separable continuous field C*-algebra over $X$ with all fibers isomorphic to the same stable Kirchberg algebra $D$. We show that if $D$ has finitely generated K-theory and it satisfies the Universal Coefficient Theorem in KK-theory, then there exists a dense open subset $U$ of $X$ such that the ideal $A(U)$ is locally trivial. The assumptions that the space $X$ is finite dimensional and that the K-theory of the fiber is finitely generated are necessary.

R. Antoine, J.Bosa, F. Perera and H. Petzka, "Geometric structure of dimension functions on certain continuous fields", J. Funct. Anal., 266 (2014), no.4, pp. 2403-2423. [pdf]

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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.

R. Antoine, J.Bosa and F. Perera, "The Cuntz semigroup of Continuous Fields", Indiana Univ. Math. J. 62 (2013), no.4, pp. 1105-1131. [pdf]

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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.

R. Antoine, J.Bosa and F. Perera, "Completions of monoids with applications to the Cuntz Semigroup", in International Journal of Mathematics, 22 (6) (2011), pp. 837-861. [pdf]

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We provide an abstract categorical framework that relates the Cuntz semigroups of the C*-algebras $A$ and $A\otimes K$. 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*-gebras.

PhD Thesis:


Continuous Fields of C*-algebras, their Cuntz Semigroup and the Geometry of Dimension Functions, Universitat Autònoma de Barcelona, 26 September 2013. [pdf of Thesis] [Presentation] Supervisor: Francesc Perera (UAB).

Msc Thesis:


Complecions del semigrup de Cuntz, Universitat Autònoma de Barcelona, September 2009. Supervisor: Francesc Perera (UAB).

Current:


Past:


Alvaro Sanchez. Joint PhD-student with Prof. P. Ara at Universitat Autònoma de Barcelona.
(2017 - 2022)

Dr. Gabriele Tornetta. Joint PhD-student with Dr. J. Zacharias at University of Glasgow.
(2013 - 2016)

Talks

Advanced Courses


"The Cuntz semigroup: Classification of C*-algebras", a series lectures of 5 hours presented at "The 4rth Workshop on Operator Algebras and its applications", School of Mathematics, IPM, Tehran, Iran. December 2016

Comunications


"Classification of C*-algebras. History and Final Results", presented at joint meeting RSME, SCM and Polish Math Society, University of Lodz (Poland) September 2023

"Almost Elementary C*-algebras", presented at workshop C*-days 2023, University of Prague (Czech Republic) May 2023

"Stable Elements and property (S)", presented at workshop Cuntz Semigroup 2022, University of Kiel (Germany) October 2022

"Almost Elementary Dynamical Systems", presented at workshop C*- algebras: Structure and dynamics (Sde Boker (Israel)) May. 2022

"The Realization problem for von Neumann regular rings", presented at Zagreb Workshop on Operator Algebras (on-line). June 2020

"Adaptable Separated graphs and Realization Problem", presented at Advanced Course on Geometry, Topology and Algebra, CRM, Bellaterra, Spain. May 2019

"Nuclear dimension of O_\infty - stable algebras", presented at School I-Operator algebras, Groups and applications to Quatum Informatiton, ICMAT, Madrid, Spain. March 2019

"Classification of C*-algebras and Nuclear dimension of O_\infty - stable algebras", presented at Joint meeting Spain- Brazil in Mathematics, Cadiz, Spain. December 2018.

"Villadsen Algebras and some regularity conditions", presented at Masterclass: Applications of the UCT for C*-algebras, University of Copenhagen, Copenhagen. October 2017.

"The dynamical Cuntz Semigroup", presented at Young Mathematicians working on C*-algebras, University of Copenhagen, Copenhagen. August 2017.

"The dynamical Cuntz Semigroup", presented at Workshop C*-algebras and Dynamical Systems", Centre de Recerca de Matemàtiques, Barcelona. March 2017.

"Classification of C*-algebras. Comparison Properties of the Cuntz semigroup and applications to C*-algebras", presented at Special weeek on Operator Algebras, Research Center of Op. Algebras, East China Normal University, Shanghai, China. June 2016.

"Comparison properties on Cu-semigroups, with applications to C*-algebras", presented in "Workshop Classification and Dynamical systems I:C*-algebras", Mittag-Leffler Institute, Stockholm. February 2016.

"Covering Dimension and 2-coloured classification", presented in "Edinburgh Math. Soc.-Soc. Catalana de Mat. joint meeting", Barcelona. May 2015.

"Equivalent definitions of the Corona Factorization Property", presented in "Workshop on C*-algebras: Structure and Classification", University of Muenster. April 2015.

"Nuclear Dimension of Kirchberg Algebras", presented in "Conference of Classification, Structure, Amenability and Regularity (CSTAR)", University of Glasgow. September 2014.

"The Category Cu. Which maps are the correct ones? *-homomorphisms or cpc order zero maps?", presented in "Workshop on C*-algebras and Dynamical Systems" Fields Institute of Toronto. June 2014.

"The Category Cu. Which maps are the correct ones? *-homomorphisms or cpc order zero maps?", presented in "Scottish Operator Algebras Seminar", University of Glasgow. March 2014.

"Local triviality for Continuous Fields of C*-algebras", presented in "School and Workshop: Topics in Operator Algebras and Applications" ICMAT-International Ciencias Matemáticas. September 2013.

"Local triviality of Continuous Fields",presented in "The structure and classification of nuclear C*-algebras" ICMS-International Centre for Mathematical Sciences April 2013.

"Dimension functions on C(X,A)", presented in "Wabash Modern Analysis Mini-conference" Indiana University-Purdue University. September 2012.

"The Cuntz semigroup of Continuous Fields", presented in "Jornadas Españolas de Teoríax de Anillos" Universidad de Granada. June 2012.

"Dimension functions on C*-algebras", presented in "Master Class on the Nuclear Dimension of C*-algebras" University of Copenhaguen. November 2011.

"The Cuntz semigroup of certain C(X)-algebra",presented in "Conference on Structure and Classification of C*-algebras" CRM-Centre Recerca Matemàtica, Bellaterra. June 2011.

"The Cuntz semigroup of a C(X)-algebra",presented in "Great Plains Operator Theory Symposium 2011" University of Phoenix. May 2011.

"C*-algebras and their classification", presented in "Jornada de Joves Matemàtics de la Societat Catalana de Matemàtiques (SCM)" Barcelona. November 2010.

"Classification of Monoids. Applications to the Cuntz semigroup", presented in "Master Class on Classification of C*-algebras", University of Copenhaguen. November 2009.

Seminar Talks


"The Realization problem for von Neumann regular rings", Algebra and Number Theory seminar, presented at University of Graz. June 2022

"Clasificación de C*-álgebraS sep. simples unitales y nucleares. Historia y resultados importantes", Seminario Rubio de Francia, presented at Universidad de Zaragoza. Feb. 2022

"El Problema de Realització i Grafs separats adaptables", Seminari G i C, presented at Universitat de Lleida. Jul. 2021

"Stable Elements and Property(S)", Seminari de Teoria d'Anells, presented at Universitat Autònoma de Barcelona. May 2021

"The Realization problem for von Neumann regular rings", Abend Seminars on Graph Algebras, presented at University of West Sydney (on-line). June 2020

"Classification of C*-algebras", GAMT seminar, presented at University of Cardiff October 2019.

"Realization for Von Neumann regular rings", Operator Algebra seminar, presented at University of Trondheim June 2019.

"Adaptable Separated Graphs", Operator Algebra seminar, presented at University of Glasgow, January 2019.

"Representation problem for Von Neumann regular rings ", Ring Theory seminar, presented at Universitat Autònoma de Barcelona, June 2018.

"Representation problem for Von Neumann regular rings and Steinberg Algebras", Operator Algebra seminar, presented at University of Muenster, May 2018.

"The representation problem for Von Neumann regular rings", Operator Algebra seminar, presented at University of Glasgow, February 2018.

"Dynamical Cuntz Semigroup", Operator Algebra seminar during IRP Operator Algebras, presented at Centre de Recerca Matemàtica, June 2017.

"Comparison Properties of the Cuntz Semigroup and applications to C*-algebras", Algebra seminar, Universitat Autònoma de Barcelona, May 2016.

"Classification of C*-algebras and the Category Cu", Operator Algebra seminar, University of Wollongong, October 2015.

"Classification of maps by traces", Operator Algebra seminar, Newcastle University, February 2015.

"Classificationof maps by traces", Operator Algebra seminar, University of Muenster, March 2014.

"The Category Cu. Which maps are the correct ones? *-homomorphisms or cpc order zero maps?", Geometry-Analysis seminar, Queen Mary-University of London, March 2014.

"The Cuntz semigroup of some Continuous Fields, Operator algebra seminar, University of Glasgow, December 2013.

"The Cuntz semigroup and Dimension Functions on C(X,A), Operator algebra seminar, Purdue University, U.S.A., October 2012.

"The Cuntz semigroup of certain C(X)-algebras", Operator algebra talk, CRM-Centre de Recerca Matemàtica, Bellaterr, May 2011.

"Completions of monoids. Applications to the Cuntz Semigroup", Operator algebra talk,Department of Mathematical Sciences-University of Copenhaguen, February 2010.

"Complecions de Semigrups: Aplicacions al Semigrup de Cuntz (II)", Algebra seminar, Departament de Matemàtiques-Universitat Autònoma de Barcelona, October 2009.

"Complecions de Semigrups: Aplicacions al Semigrup de Cuntz (I)", Algebra seminar, Departament de Matemàtiques-Universitat Autònoma de Barcelona, October 2009.

Organized Scientific Events

With S. Goldstein, B. Kwaniewski, F. Perera, M. Wasilewski organized the “OA-session in the Spanish-Polish Mathematical meeting”, Lodz, Poland. Sep. 2023

With R. Antoine and F. Perera organized the Conference " III Barcelona Weekend on Operator Algebras and Noncommutative Algebra", Centre de Recerca Matemàtica, Bellaterra, Jan. 2023

With F. Perera organized the Conference " II Barcelona Weekend on Operator Algebras and Noncommutative Algebra", Centre de Recerca Matemàtica, Bellaterra, Sep. 2019.

With M. Gonchenko and M. Sanz-sole organized the Conference "2nd BMS-BGSmath Junior Meeting", TU Berline and Institute Berlin (ZIB), June 2019.

With F. Perera organized the Conference "Barcelona Weekend on Operator Algebras", Centre de Recerca Matemàtica, Bellaterra, Nov. 2018.

Organizer of "Operator Rings Theory seminars", at Universitat Autònoma de Barcelona since Sep 2018.

Organizer of "Operator Algebra seminars", at University of Glasgow 2014.

With S. White, W. Winter and J. Zacharias organized the Conference "Classification, Structure, Amenability and Regularity (CSTAR)", University of Glasgow, 1-5 Set. 2014.

With S. White, W. Winter and J. Zacharias organized the Master Class "Classification, Structure, Amenability and Regularity (CSTAR)", University of Glasgow, 21-29 May 2014.

With Ramon Antoine, Pere Ara and Francesc Perera organized the Workshop "Workshop on Dynamics and C*-algebras", in CRM-Centre Recerca Matemàtica, Bellaterra, 6-8 April , 2011.

With Ramon Antoine, Pere Ara and Francesc Perera, organized the Conference "Structure and Classification of C*-algebras", in CRM-Centre Recerca Matemàtica, Bellaterra, 6-10 June , 2011.

With Ramon Antoine, Pere Ara and Francesc Perera, organized the Advanced course"Structure and Classification of C*-algebras", in CRM-Centre Recerca Matemàtica, Bellaterra, 6-10 June , 2011.