Categoria: Operator Algebras

Laurent Cantier (UAB-Czech Academy of Sciences)

Webbing transformations and C*-algebras

Abstract: In the recent light of the emergence of new invariants for non-simple C*-algebras, we expose a categorical construction that we refer to as the webbing transformation, allowing to generically merge distinct C*-invariants together. E.g. the Cuntz semigroup together with K-theoretical data. One of the benefits is to naturally incorporate the data encoded within any (closed two-sided) ideals. In this talk, we will first define our categorical framework and study properties of these webbed objects, including an ideal-quotient theory, to then venture into their possible impact on the classification of non-simple C*-algebras.

Fernando Lledó (UC·M-ICMAT)

Finite dimensional approximations in two classes of operator algebras

Abstract: In this talk I will present finite dimensional matrix approximations in two classes of operator algebras.

Eduard Vilalta (Universitat Autònoma de Barcelona)

Nowhere scattered multiplier algebras

Abstract: A natural assumption that ensures sufficient noncommutativity of a C*-algebra is nowhere scatteredness, which in one of its many formulations asks the algebra to contain no nonzero elementary ideal-quotients. This notion enjoys many good permanence properties, but fails to pass to certain unitizations. For example, no minimal unitization of a non-unital C*-algebra (nowhere scattered or not) can ever be nowhere scattered. However, it is unclear when a nowhere scattered C*-algebra has a nowhere scattered multiplier algebra.

In this talk, I will give sufficient conditions under which this happens. It will follow from the main result of the talk that a $\sigma$-unital C*-algebra of finite nuclear dimension, or of real rank zero, or of stable rank one and k-comparison, is nowhere scattered if and only if its multiplier algebra is. I will also give some examples of nowhere scattered C*-algebras whose multiplier algebra is not nowhere scattered.

Joachim Zacharias (University of Glasgow)

On a finite section method to approximate exact C*-algebras

Abstract: Exact C*-algebras are an important class of C*-algebras which is closed under subalgebras and contains all nuclear C*-algebras. A basic result due to Kirchberg asserts that any such separable C*-algebra is a sub-quotient of a UHF-algebra.

We give a short survey on exact C*-algebras, indicating a simplified ‘finite-section’ approach to Kirchberg’s basic result and outline possible applications, including a Stone-Weierstrass type Theorem for exact C*-algebras.

Francesc Perera (Universitat Autònoma de Barcelona)

The dynamical Cuntz semigroup and crossed products

Abstract: In this talk I shall discuss the definition of dynamical subequivalence for open subsets of a compact topological space and its natural counterpart involving Cuntz subequivalence. This will lead to the definition of the dynamical Cuntz semigroup. I will mention how this semigroup is related to the construction of crossed products in various categories. This is part of joint work with J. Bosa, J. Wu, and J. Zacharias, and also R. Antoine and H. Thiel.