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.

Scheme theoretic tropicalization

In this paper, we introduce ordered blueprints and ordered blue schemes, which serve as a common language for the different approaches to tropicalizations and which enhances tropical varieties with a schematic structure. As an abstract concept, we consider a tropicalization as a moduli problem about extensions of a given valuation $v:k\to T$ between ordered blueprints $k$ and $T$. If $T$ is idempotent, then we show that a generalization of the Giansiracusa bend relation leads to a representing object for the tropicalization, and that it has yet another interpretation in terms of a base change along $v$. We call such a representing object a scheme theoretic tropicalization. This theory recovers and improves other approaches to tropicalizations as we explain with care in the second part of this text. The Berkovich analytification and the Kajiwara-Payne tropicalization appear as rational point sets of a scheme theoretic tropicalization. The same holds true for its generalization by Foster and Ranganathan to higher rank valuations. The scheme theoretic Giansiracusa tropicalization can be recovered from the scheme theoretic tropicalizations in our sense. We obtain an improvement due to the resulting blueprint structure, which is sufficient to remember the Maclagan-Rinc\'on weights. The Macpherson analytification has an interpretation in terms of a scheme theoretic tropicalization, and we give an alternative approach to Macpherson's construction of tropicalizations. The Thuillier analytification and the Ulirsch tropicalization are rational point sets of a scheme theoretic tropicalization. Our approach yields a generalization to any, possibly nontrivial, valuation $v:k\to T$ with idempotent $T$ and enhances the tropicalization with a schematic structure.Comment: 66 pages; for information about the changes in this version of the paper, please cf. the paragraph "Differences to previous versions" in the introductio

The geometry of blueprints. Part I: Algebraic background and scheme theory

In this paper, we introduce the category of blueprints, which is a category of algebraic objects that include both commutative (semi)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp.\ congruences for rings and monoids and leads to a common scheme theory. In particular, it bridges the gap between usual schemes and $\mathbb{F}_1$-schemes (after Kato, Deitmar and Connes-Consani). Beside this unification, the category of blueprints contains new interesting objects as "improved" cyclotomic field extensions $\mathbb{F}_{1^n}$ of $\mathbb{F}_1$ and "archimedean valuation rings". It also yields a notion of semiring schemes. This first paper lays the foundation for subsequent projects, which are devoted to the following problems: Tits' idea of Chevalley groups over $\mathbb{F}_1$, congruence schemes, sheaf cohomology, $K$-theory and a unified view on analytic geometry over $\mathbb{F}_1$, adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry.Comment: Slightly revised and extended version as in print. 51 page

Algebraic structures of tropical mathematics

Tropical mathematics often is defined over an ordered cancellative monoid \tM, usually taken to be (\RR, +) or (\QQ, +). Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted algebraic structure theory, and also do not reflect certain valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques. In this paper we describe an alternative structure, more compatible with valuation theory, studied by the authors over the past few years, that permits fuller use of algebraic theory especially in understanding the underlying tropical geometry. The idempotent max-plus algebra $A$ of an ordered monoid \tM is replaced by R: = L\times \tM, where $L$ is a given indexing semiring (not necessarily with 0). In this case we say $R$ layered by $L$. When $L$ is trivial, i.e, $L=\{1\}$, $R$ is the usual bipotent max-plus algebra. When $L=\{1,\infty\}$ we recover the "standard" supertropical structure with its "ghost" layer. When L = \NN we can describe multiple roots of polynomials via a "layering function" $s: R \to L$. Likewise, one can define the layering $s: R^{(n)} \to L^{(n)}$ componentwise; vectors $v_1, \dots, v_m$ are called tropically dependent if each component of some nontrivial linear combination \sum \a_i v_i is a ghost, for "tangible" \a_i \in R. Then an $n\times n$ matrix has tropically dependent rows iff its permanent is a ghost. We explain how supertropical algebras, and more generally layered algebras, provide a robust algebraic foundation for tropical linear algebra, in which many classical tools are available. In the process, we provide some new results concerning the rank of d-independent sets (such as the fact that they are semi-additive),put them in the context of supertropical bilinear forms, and lay the matrix theory in the framework of identities of semirings.Comment: 19 page

Provenance for Aggregate Queries

We study in this paper provenance information for queries with aggregation. Provenance information was studied in the context of various query languages that do not allow for aggregation, and recent work has suggested to capture provenance by annotating the different database tuples with elements of a commutative semiring and propagating the annotations through query evaluation. We show that aggregate queries pose novel challenges rendering this approach inapplicable. Consequently, we propose a new approach, where we annotate with provenance information not just tuples but also the individual values within tuples, using provenance to describe the values computation. We realize this approach in a concrete construction, first for "simple" queries where the aggregation operator is the last one applied, and then for arbitrary (positive) relational algebra queries with aggregation; the latter queries are shown to be more challenging in this context. Finally, we use aggregation to encode queries with difference, and study the semantics obtained for such queries on provenance annotated databases