Tensor products and regularity properties of Cuntz semigroups

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 $Cu(A\otimes B)$ 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.Comment: 195 pages; revised version; several proofs streamlined; some results corrected, in particular added 5.2.3-5.2.

Generalized comaximal factorization of ideals

AbstractWe generalize the notion of comaximal factorization of ring ideals to the language of weak ideal systems on monoids and prove several results generalizing and extending previous work. We also develop some topological methods for dealing with comaximal factorization and some related finitary weak ideal system problems

Ultrametric spaces of branches on arborescent singularities

Let $S$ be a normal complex analytic surface singularity. We say that $S$ is arborescent if the dual graph of any resolution of it is a tree. Whenever $A,B$ are distinct branches on $S$, we denote by $A \cdot B$ their intersection number in the sense of Mumford. If $L$ is a fixed branch, we define $U_L(A,B)= (L \cdot A)(L \cdot B)(A \cdot B)^{-1}$ when $A \neq B$ and $U_L(A,A) =0$ otherwise. We generalize a theorem of P{\l}oski concerning smooth germs of surfaces, by proving that whenever $S$ is arborescent, then $U_L$ is an ultrametric on the set of branches of $S$ different from $L$. We compute the maximum of $U_L$, which gives an analog of a theorem of Teissier. We show that $U_L$ encodes topological information about the structure of the embedded resolutions of any finite set of branches. This generalizes a theorem of Favre and Jonsson concerning the case when both $S$ and $L$ are smooth. We generalize also from smooth germs to arbitrary arborescent ones their valuative interpretation of the dual trees of the resolutions of $S$. Our proofs are based in an essential way on a determinantal identity of Eisenbud and Neumann.Comment: 37 pages, 16 figures. Compared to the first version on Arxiv, il has a new section 4.3, accompanied by 2 new figures. Several passages were clarified and the typos discovered in the meantime were correcte

Categorical Operational Physics

Many insights into the quantum world can be found by studying it from amongst more general operational theories of physics. In this thesis, we develop an approach to the study of such theories purely in terms of the behaviour of their processes, as described mathematically through the language of category theory. This extends a framework for quantum processes known as categorical quantum mechanics (CQM) due to Abramsky and Coecke. We first consider categorical frameworks for operational theories. We introduce a notion of such theory, based on those of Chiribella, D'Ariano and Perinotti (CDP), but more general than the probabilistic ones typically considered. We establish a correspondence between these and what we call "operational categories", using features introduced by Jacobs et al. in effectus theory, an area of categorical logic to which we provide an operational interpretation. We then see how to pass to a broader category of "super-causal" processes, allowing for the powerful diagrammatic features of CQM. Next we study operational theories themselves. We survey numerous principles that a theory may satisfy, treating them in a basic diagrammatic setting, and relating notions from probabilistic theories, CQM and effectus theory. We provide a new description of superpositions in the category of pure quantum processes, using this to give an abstract construction of the category of Hilbert spaces and linear maps. Finally, we reconstruct finite-dimensional quantum theory itself. More broadly, we give a recipe for recovering a class of generalised quantum theories, before instantiating it with operational principles inspired by an earlier reconstruction due to CDP. This reconstruction is fully categorical, not requiring the usual technical assumptions of probabilistic theories. Specialising to such theories recovers both standard quantum theory and that over real Hilbert spaces.Comment: DPhil Thesis, University of Oxfor

Dyadic harmonic analysis: Non-doubling and noncommutative aspects

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 26-05-2015La presente tesis contiene resultados sobre análisis armónico diádico en distintos contextos; proporcionando estimaciones a priori para modelos diádicos de integrales singulares. La exposición de los resultados se divide en tres partes. En la primera se caracterizan las medidas de Borel en R para las cuales la transformada de Hilbert diádica asociada es de tipo débil (1, 1). Sorprendentemente, la clase de medidas obtenida contiene estrictamente a las medidas diádicamente doblantes y está contenida estrictamente en la clase de Borel. Se demuestra además que la clase dual caracteriza el tipo débil (1, 1) del adjunto de la transformada de Hilbert diádica. La herramienta principal es una nueva descomposición de Calderón- Zygmund válida para medidas de Borel generales y de interés independiente. Caracterizaciones análogas del tipo débil (1, 1) para operadores Haar shift multidimensionales son obtenidas en términos de dos sistemas de Haar generalizados y no necesariamente cancelativos. Los paraproductos diádicos y sus adjuntos figuran como casos particulares importantes. Por otro lado, es bien sabido que operadores de Calderón-Zygmund con núcleos matriciales — incluso aquellos con buenas propiedades de tamaño y suavidad o cancelación — carecen de estimaciones en Lp semiconmutativas para p 6= 2. En la segunda parte de la tesis se obtienen estimaciones de tipo débil (1, 1) de operadores perfectamente diádicos y, en general para operadores Haar shift, en términos de una descomposición fila/columna de la función de partida. Se muestra también que operadores de Calderón-Zygmund generales satisfacen estimaciones de tipo H1 ! L1, que junto con estimaciones de tipo L1 ! BMO, implican estimaciones fila/columna en espacios Lp semiconmutativos. El enfoque presentado es aplicable a transformadas de martingala y paraproductos con símbolos no conmutativos, para los que obtenemos estimaciones análogas. La tercera parte está dedicada a la generalización semiconmutativa de los resultados obtenidos en la primera parte. Esto es, a la caracterización del tipo débil (1, 1) de operadores Haar shift definidos en términos de dos sistemas de Haar generalizados adaptados a una medida de Borel y con símbolos conmutativos. Así como en el caso conmutativo, el principal recurso técnico es una versión no conmutativa de la descomposición de Calderón-Zygmund introducida en la primera parteThis thesis is divided into three parts, each presenting results on dyadic harmonic analysis in different settings. More specifically, it provides a priori estimates of dyadic and singular integral operators in the non-doubling and semicommutative frameworks. In Part I we characterize the locally finite Borel measures μ on R for which the associated dyadic Hilbert transform satisfy L1(μ) ! L1,1(μ) estimates. Surprisingly, the class of such measures is strictly bigger than the standard class of dyadically doubling measures and strictly smaller than the whole Borel class. We further show that a dual class characterizes the weak-type (1, 1) of the adjoint of the dyadic Hilbert transform. In higher dimensions, we provide a complete characterization of the weak-type (1, 1) of arbitrary Haar shift operators —cancellative or not—written in terms of two generalized Haar systems, including dyadic paraproducts. The main tool used in Part I is a new Calderón-Zygmund decomposition valid for arbitrary Borel measures which is of independent interest. On the other hand, it is well known that Calderón-Zygmund operators with noncommuting kernels may fail to be Lp bounded in semicommutative Lp spaces for p 6= 2, even for kernels with good size and smoothness properties or having dyadic cancellation properties. In Part II we obtain weak-type (1, 1) estimates for perfect dyadic Calderón-Zygmund operators associated to noncommuting kernels in terms of a row/column decomposition of the input function. Analogous estimates are also proved for arbitrary Haar shift operators. General Calderón-Zygmund operators satisfy H1 ! L1 type estimates. In conjunction with L1 ! BMO type estimates, we get similar row/column Lp estimates. The approach here presented also applies to martingale transforms and paraproducts with noncommuting symbols for which we obtain analogous estimates. In Part III we obtain a complete characterization of the weak-type (1, 1) of commuting Haar shift operators in terms of generalized Haar systems adapted to a Borel measure μ in the semicommutative setting. The main technical tool in our method is a noncommutative Calderón-Zygmund decomposition that generalizes the Calderón-Zygmund decomposition used in the first part

The modern theory of Cuntz semigroups of C*-algebras

We give a detailed introduction to the theory of Cuntz semigroups for C*-algebras. Beginning with the most basic definitions and technical lemmas, we present several results of historical importance, such as Cuntz's theorem on the existence of quasitraces, R{\o}rdam's proof that $\mathcal{Z}$-stability implies strict comparison, and Toms' example of a non $\mathcal{Z}$-stable simple, nuclear C*-algebra. We also give the reader an extensive overview of the state of the art and the modern approach to the theory, including the recent results for C*-algebras of stable rank one (for example, the Blackadar-Handelman conjecture and the realization of ranks), as well as the abstract study of the Cuntz category $\mathbf{Cu}$.Comment: 61 pages. V2: minor changes, fixed typos, added references and acknowledgements. 62 page

Quantum Statistical Mechanics, L-Series and Anabelian Geometry I: Partition Functions

The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical (QSM) system, built from abelian class field theory. We introduce a general notion of isomorphism of QSM-systems and prove that it preserves (extremal) KMS equilibrium states. We prove that two number fields with isomorphic quantum statistical mechanical systems are arithmetically equivalent, i.e., have the same zeta function. If one of the fields is normal over ℚ, this implies that the fields are isomorphic. Thus, in this case, isomorphism of QSM-systems is the same as isomorphism of number fields, and the noncommutative space built from the abelianized Galois group can replace the anabelian absolute Galois group from the theorem of Neukirch and Uchida. This paper is an updated version of part of [9]. We have split the original preprint into various parts, depending on the methods that are used in them. In the current part, these belong mainly to mathematical physics

Skeleta in non-Archimedean and tropical geometry

I describe an algebro-geometric theory of skeleta, which provides a unified setting for the study of tropical varieties, skeleta of non-Archimedean analytic spaces, and affine manifolds with singularities. Skeleta are spaces equipped with a structure sheaf of topological semirings, and are locally modelled on the spectra of the same. The primary result of this paper is that the topological space X underlying a non-Archimedean analytic space may locally be recovered from the sheaf |∂x| of pointwise valuations' of its analytic functions in other words, (X,|∂x|) is a skeleton.Open Acces