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.