Related papers: The Cuntz semigroup and domain theory
We present a $C^*$-algebra which is naturally associated to the $ax+b$-semigroup over $\mathbb N$. It is simple and purely infinite and can be obtained from the algebra considered by Bost and Connes by adding one unitary generator which…
The notion of extension of a given $C^*$-category $C$ by a $C^*$-algebra $A$ is introduced. In the commutative case $A = C(\Omega)$, the objects of the extension category are interpreted as fiber bundles over $\Omega$ of objects belonging…
A C*-algebra is determined to a great extent by the partial order of its commutative C*-algebras. We study order-theoretic properties of this dcpo. Many properties coincide: the dcpo is, equivalently, algebraic, continuous, meet-continuous,…
Recently, Cuntz and Li introduced the C^*-algebra A[R] associated to an integral domain R with finite quotients. In this paper, we show that A[R] is a partial group algebra of the group $K \rtimes K^x$ with suitable relations, where K is…
In a recent paper, Pardo and the first named author introduced a class of C*-algebras which which are constructed from an action of a group on a graph. This class was shown to include many C*-algebras of interest, including all Kirchberg…
We calculate the Cuntz semigroup of the tensor product A with A. We restrict our attention to C*-algebras A which are unital, simple, nuclear, stably finite, have stable rank one, absorbs the Jiang-Su algebra tensorially and satisfy the…
A duality is discussed for Lie group bundles vs. certain tensor categories with non-simple identity, in the setting of Nistor-Troitsky gauge-equivariant K-theory. As an application, we study C*-algebra bundles with fibre a fixed-point…
In prior work we described how the Cuntz-Pimsner construction may be viewed as a functor. The domain of this functor is a category whose objects are $C^*$-correspondences and morphisms are isomorphism classes of certain pairs comprised of a…
For any given Cuntz semigroup, we introduce a function associated to it, called the ``Rank Ratio Function". This function ensures global control over the oscillation of the rank of any pair of elements in the Cuntz semigroup and, also,…
The problem of expressing a selfadjoint element that is zero on every bounded trace as a finite sum (or a limit of sums) of commutators is investigated in the setting of C*-algebras of finite nuclear dimension. Upper bounds -- in terms of…
We prove a double commutant theorem for separable subalgebras of a wide class of corona C*-algebras, largely resolving a problem posed by Pedersen. Double commutant theorems originated with von Neumann, whose seminal result evolved into an…
In this paper which is the first of a series of papers on smooth structures, the concepts of C-structures and smooth structures are introduced and studied. The notion of smooth structure on semi-integral domains is given. It is shown that…
The aim of this project is to attach a geometric structure to the ring of integers. It is generally assumed that the spectrum $\mathrm{Spec}(\mathbb{Z})$ defined by Grothendieck serves this purpose. However, it is still not clear what…
It is shown in this paper that two positive elements of a C*-algebra agree on all lower semicontinuous traces if and only if they are equivalent in the sense of Cuntz and Pedersen. A similar result is also obtained in the more general case…
While there is only one natural dimension concept for separable, metric spaces, the theory of dimension in noncommutative topology ramifies into different important concepts. To accommodate this, we introduce the abstract notion of a…
The Cuntz algebra carries in a natural way the structure of a module algebra over the quantized universal enveloping algebra $U_q(g)$, and the structure of a co-module algebra over the quantum group $G_q$ associated with $U_q(g)$. These two…
The notion and theory of the quantum space of all maps from a quantum space pioneered by So{\l}tan have been mainly focused on finite-dimensional C*-algebras which are matrix algebra bundles over a finite set $S$. We propose a modification…
We introduce a Fra\"iss\'e theory for abstract Cuntz semigroups akin to the theory of Fra\"iss\'e categories developed by Kubi\'s. In particular, we show that any (Cuntz) Fra\"iss\'e category has a unique Fra\"iss\'e limit which is both…
We prove a version of uniqueness theorem for Cuntz-Pimsner algebras of discrete product systems over semigroups of Ore type. To this end, we introduce Doplicher-Roberts picture of Cuntz-Pimsner algebras, and the semigroup dual to a product…
Traditional knowledge graphs are constrained by fixed ontologies that organize concepts within rigid hierarchical structures. The root cause lies in treating domains as implicit context rather than as explicit, reasoning-level components.…