Recent developments within the area of model theory for -algebras have led to manyinteresting questions and remarkable results. One such question, posed by Farah et al. in 2021, is the following: is the Jiang-Su algebra existentially closed in the class of unital, nuclear, stably finite, projectionless -algebras? This question can be rephrased in terms of existence of an embedding into the ultrapower of Z, which itself can be seen as a -algebraic analogue of the famous Connes embedding problem for von Neumann algebras. This prompts the study of when such an embedding exists. In joint work with Jennifer Pi (University of Oxford), we show that cones over separable -algebras always embed into the ultrapower of , a direct analogue of Voiculescu’s result that such  -algebras are quasidiagonal. As such, we propose embeddability into the ultrapower of as a similar regularity property. This is supported by our main result, which establishes homotopy invariance of this property under some additional hypotheses. In this talk, I will discuss these results, as well as their connections with model theoretic and  -algebraic problems.

AF-embeddability, ie the question whether a given -algebra can be realised as a subalgebra of an AF-algebra has been studied for a long time with prominent early results by Pimsner and Voicuescu who constructed such embeddings for irrational rotation algebras in 1980. Since then many AF-embeddings have been constructed for concrete examples but also many non-constructive AF-embeddability results have been obtained for classes of algebras typically assuming the UCT. In this talk we consider a separable unital -algebra A of decomposition rank at most 1 and construct from a suitable system of 1-decomposable cpc-approximations an AF-algebra E together with an embedding of A into E and a conditional expectation of E onto A without assuming the UCT and indicate some applications and related results. 

For any ring R a new invariant has been defined in the form of a partially ordered abelian monoid, built from an equivalence relation on the class of countably generated projective modules and denoted by S(R). It has been named the Cuntz semigroup of the ring R, since its construction is akin to the manufacture of the Cuntz semigroup of a -algebra using countably generated Hilbert modules. Two questions about this semigroup that remain open in the general case are whether S(R) belongs to the abstract Cuntz category, Cu, and whether, given a C*-algebra A, its Cuntz semigroup as a ring and as a -algebra coincide.
In this talk I will focus on the class of rings R over which every projective module is a direct sum of finitely generated modules. I will give a representation of S(R) as a certain monoid of intervals of V (R), and I will answer the two questions mentioned. We will also compute the Cuntz semigroup for rings of continuous functions on one-dimensional spaces. As an application, for such rings we can characterize the class of a countably generated projective ideal by its trace ideal. We will see that the situation is different when considering real-valued or complex-valued functions. I will also present the related notion of left normality for a ring and see whether is satisfied for such rings of continuous functions.
This is work of my Ph.D. under the supervision of Pere Ara and Francesc Perera.

In this talk I will present a first method to construct KMS states for certain separated graph -algebras (E, C). We will consider finite and bipartite separated graphs (E, C) and base our construction on the analysis of the so-called tame algebra (E, C), a quotient of A that admits a realization as a crossed product of continuous functions on a configuration space by a partial action of a free group. The existence of KMS-states on A is obtained by constructing a measure on the configuration space that is quasi-invariant with respect to the coloured dynamics. I will also mention during the talk concrete examples of the different structures involved.
This is joint work with P. Ara (UAB), J. Claramunt (UPC) and E. Gillaspy (U. Montana)

In this talk I will present a second method to construct KMS states for general separated graph -algebras . I will start by introducing the main tool utilized, namely amalgamated probability, a generalization of free probability, and how it will play a central role in the search for KMS states. Necessary conditions for existence of KMS states on will be given and, under mild conditions, we prove that such conditions are also suffcient. I will demonstrate, in specific examples, how to use those conditions to explicitly construct KMS states on , which in general may differ from the KMS states constructed using the first method presented by F. Lledó. This shows that, in general, the KMS simplex of separated graph -algebras is much richer than the usual graph -algebras.

This is joint work with Pere Ara (UAB), Elizabeth Gillaspy (UM) and Fernando Lledó (UC3M).

Strict comparison is a fundamental property of C∗-algebras, originally introduced by Blackadar to capture the appropiate generalization of Murray-von Neumann comparison theory for factors. Since its introduction, it has become a cornerstone of the modern structure theory of C∗-algebras.
A key feature of strict comparison is its versatility: it is useful for both nuclear and non-nuclear C*-algebras, with applications ranging from classification results to the recent negative solution of the Tarski problem. A central line of research has been to determine which (reduced, possibly twisted) group C*-algebras possess this property—a question that has seen significant progress in both the amenable and non-amenable settings.
In this talk, I will outline the main ideas underlying these developments and discuss, in particular, recent joint work with S. Raum and H. Thiel

A groups is called coherent if it has the property that any finitely generated subgroup is also finitely presented. Coherence of groups is a very interesting phenomenon that one encounters mainly in small dimension. In the case of a right angled Artin group, it is a classical result by Droms that coherence can be characterized in terms of the defining graph and it was shown later by Droms, B. Servatius and H. Servatius that it is equivalent to the derived group being free. The characterization in terms of the defining graph has been extended to the family of Artin groups by Gordon and Wise and in this talk we will show that for arbitrary Artin groups the same characterization of coherence in terms of the derived group holds true. We will also review some recent results on coherence and discuss the relationship with properties such as the coherence of the group ring, a relationship which is far from being well understood in general.

I will talk mostly about cone C*-algebras. We will discuss their properties- some known (e.g. quasidiagonality and quasidiagonality of amenable traces) and some new. We also will discuss homotopy invariance of several C*-algebraic properties.