Related papers: Semifinite spectral triples associated with graph …
We investigate conditions on a graph $C^*$-algebra for the existence of a faithful semifinite trace. Using such a trace and the natural gauge action of the circle on the graph algebra, we construct a smooth $(1,\infty)$-summable semfinite…
Recently, examples of an index theory for KMS states of circle actions were discovered, \cite{CPR2,CRT}. We show that these examples are not isolated. Rather there is a general framework in which we use KMS states for circle actions on a…
Inspired by results for graph $C^*$-algebras, we investigate connections between the ideal structure of an inverse semigroup $S$ and that of its tight $C^*$-algebra by relating ideals in $S$ to certain open invariant sets in the associated…
This paper is comprised of two related parts. First we discuss which k-graph algebras have faithful gauge invariant traces, where the gauge action of $\T^k$ is the canonical one. We give a sufficient condition for the existence of such a…
This survey deals with the construction of a category of spectral triples that is compatible with the Kasparov product in $KK$-theory. These notes serve as an intuitive guide to these results, avoiding the necessary technical proofs. We…
In previous papers, we constructed smooth (1,\infty)-summable semfinite spectral triples for graph algebras with a faithful trace, and (k,\infty)-summable semifinite spectral triples for k-graph algebras. In this paper we identify classes…
By introducing a notion of smooth connection for unbounded $KK$-cycles, we show that the Kasparov product of such cycles can be defined directly, by an algebraic formula. In order to achieve this it is necessary to develop a framework of…
A new construction of a semifinite spectral triple on an algebra of holonomy loops is presented. The construction is canonically associated to quantum gravity and is an alternative version of the spectral triple presented in…
We define KK-theory spectra associated to C*-categories and look at certain instances of the Kasparov product at this level. This machinery is used to give a description of the analytic assembly map as a natural map of spectra.
A semigroupoid is a set equipped with a partially defined associative operation. Given a semigroupoid \Lambda we construct a C*-algebra C*(\Lambda) from it. We then present two main examples of semigroupoids, namely the Markov semigroupoid…
Using associated trees, we construct a spectral triple for the C$^*$-algebra of continuous functions on the ring of integers $R$ of a nonarchimedean local field $F$ of characteristic zero, and investigate its properties. Remarkably, the…
We define the filtrated K-theory of a C*-algebra over a finite topological space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated K-theory. For finite spaces with…
We develop a theory of type semigroups for arbitrary twisted, not necessarily Hausdorff \'etale groupoids. The type semigroup is a dynamical version of the Cuntz semigroup. We relate it to traces, ideals, pure infiniteness, and stable…
We study semigroup C*-algebras of $ax+b$-semigroups over integral domains. The goal is to generalize several results about C*-algebras of $ax+b$-semigroups over rings of algebraic integers. We prove results concerning K-theory and…
We construct several $C^*$-algebras and spectral triples associated to the Berkovich projective line $\mathbb{P}^1_{\mathrm{Berk}}({\mathbb{C}_p})$. In the commutative setting, we construct a spectral triple as a direct limit over finite…
Using the Baum-Connes conjecture with coefficients, we develop a K-theory formula for reduced C*-algebras of strongly $0$-$E$-unitary inverse semigroups, or equivalently, for certain reduced partial crossed products. In the case of…
We introduce the notion of continuous orbit equivalence for partial dynamical systems, and give an equivalent characterization in terms of Cartan-isomorphisms for partial C*-crossed products. Both graph C*-algebras and semigroup C*-algebras…
We give a complete description of which unital graph C*-algebras are semiprojective, and use it to disprove two conjectures by Blackadar. To do so, we perform a detailed analysis of which projections are properly infinite in such…
The external Kasparov product is used to construct odd and even spectral triples on crossed products of $C^*$-algebras by actions of discrete groups which are equicontinuous in a natural sense. When the group in question is $\Z$ this gives…
We propose a definition of a modular spectral triple which covers existing examples arising from KMS-states, Podles sphere and quantum SU(2). The definition also incorporates the notion of twisted commutators appearing in recent work of…