# Operator Algebras

• #### Jueves 16

15:00 - 15:45

#### Lifts of completely positive maps

Eusebio Gardella (Universidad de Münster, Alemania)

Let $$A$$ and $$B$$ be $$C^*$$-algebras, $$A$$ separable and $$I$$ an ideal in $$B$$. We show that for any completely positive contractive linear map $$\psi\colon A\to B/I$$ there is a continuous family $$\Theta_t\colon A\to B$$, for $$t\in [1,\infty)$$, of lifts of $$\psi$$ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If $$A$$ and $$B$$ carry continuous actions of a second countable locally compact group $$G$$ such that $$I$$ is $$G$$-invariant and $$\psi$$ is equivariant, then the family $$\Theta_t$$ can be chosen to be asymptotically equivariant.

If a linear completely positive lift for $$\psi$$ exists, then we can arrange that $$\Theta_t$$ is linear and completely positive for all $$t\in [1,\infty)$$; this yields an equivariant version of the Choi-Effros lifting theorem. In the equivariant setting, if $$A$$, $$B$$ and $$\psi$$ are unital, the existence of asymptotically linear unital lifts are only guaranteed if $$G$$ is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.

This talk is based on joint work with Marzieh Forough and Klaus Thomsen.

15:45 - 16:30

#### Revisiting quantum mechanics via quantales and groupoid C*-algebras

Pedro Resende (Instituto Superior Técnico, Portugal)

It is well known that operator algebras are the mathematical language of algebraic quantum mechanics and algebraic quantum field theory. In particular, von Neumann algebras cater for a natural formalization of Heisenberg's picture'' of quantum mechanics, which rests on interpreting physical observables as being time-dependent self-adjoint operators, to which von Neumann explicitly added, still in his twenties, that physical measurements should correspond to projection operators, thus producing the first mathematically thorough account of the foundations of quantum theory. However, from a conceptual point of view deep problems remained. These, in essence, have led to the seemingly unstoppable proliferation of interpretations and modifications of quantum mechanics, despite which, to this day, the measurement problem'' remains largely unsolved. In this talk I describe ongoing work whose aim is to address this by means of a definition of space of measurements whose open sets correspond to the finite pieces of classical information that can be extracted during a measurement. The interplay between C*-algebras, locally compact étale groupoids, and quantales plays an important role. The main purpose of the talk is to explain the main ideas surrounding measurement spaces along with related facts about groupoid C*-algebras and Fell bundles.

16:45 - 17:30

#### Continuous orbit equivalence and full groups of ultragraph C*-algebras

Daniel Gonçalves (Universidade Federal de Santa Catarina, Brasil)

In this talk, we describe ultragraphs, their associated edge shift spaces (which generalize SFT for infinite alphabets), and their associated C*-algebras and groupoids. We present results regarding continuous orbit equivalence of Deaconu-Renault systems and full groups associated with groupoids. To finish, we describe how to apply these results for ultragraph algebras.

17:30 - 18:15

#### Toeplitz algebras of semigroups and their boundary quotients

Camila Senhem (Victoria University of Wellington, Nueva Zelanda)

In general there are many distinct C*-algebras generated by isometric representations of a left cancellative monoid $$P$$. Two distinguished examples are the Toeplitz C*-algebra $$\mathcal{T}_\lambda(P)$$ generated by the left regular representation of $$P$$ on $$\ell^2(P)$$, and its boundary quotient $$\partial\mathcal{T}_\lambda(P)$$. I will report on recent work with M. Laca, in which we define a universal Toeplitz C*-algebra $$\mathcal{T}_u(P)$$ for a submonoid of a group that is canonically isomorphic to Li's semigroup C*-algebra when independence holds and is the universal analogue of $$\mathcal{T}_\lambda(P)$$ also when independence fails. I plan to focus mainly on boundary quotients, explaining why the covariance algebra of the canonical product system over $$P$$ with one-dimensional fibres may be regarded as the universal analogue of $$\partial\mathcal{T}_\lambda(P)$$. I will give a concrete presentation of the full boundary quotient using a new notion of foundation sets, and give sufficient conditions on $$P$$ for $$\partial\mathcal{T}_\lambda(P)$$ to be purely infinite simple. If time permits, I will also address faithfulness of representations at the level of Toeplitz algebras.

• #### Viernes 17

15:00 - 15:45

#### Separated graphs and dynamics.

Pere Ara (Universitat Autònoma de Barcelona, España)

A separated graph is a pair $$(E,C)$$, where $$E$$ is a directed graph, $$C=\bigsqcup _{v\in E^ 0} C_v$$, and $$C_v$$ is a partition of $$r^{-1}(v)$$ (into pairwise disjoint nonempty subsets) for every vertex $$v$$. In recent years, separated graphs have been used to provide combinatorial models of several structures, often related to dynamical systems. This can be understood as a generalization of the common use of usual directed graphs in symbolic dynamics. I will survey some of these developments, including the failure of Tarski's dichotomy in the setting of topological actions, the construction of a family of ample groupoids with prescribed type semigroup, and the modeling of actions on the Cantor set.

15:45 - 16:30

#### A non-commutative topological dimension

Jorge Castillejos (Instituto de Ciencias Matemáticas ICMAT, España)

Due to the Gelfand duality, C*-algebras are regarded as non-commutative topological spaces. This approach has sparked fundamental ideas in the structure and classification theory of C*-algebras. In this talk, I will explain a notion of non-commutative covering dimension and its impact in the classification programme of C*-algebras.

16:45 - 17:30

#### Permanence properties of verbal products and verbal wreath products of groups

Román Sasyk (Universidad de Buenos Aires, Argentina)

Given a family of groups, the direct sum and the free product provide useful ways of constructing new groups out of it. Verbal products of groups, which were defined in the fifties, give many new operations in groups which share some common features with the direct sum and the free product. Arguably, these products have not been studied much in recent years and they might have been overlooked in geometric and measurable group theory.

In this talk we will review these products, give some examples and show that several group theoretical properties that are of much interest in operator algebras like amenability, Haagerup property, property (T), exactness, soficity, Connes' embedability, etc. are preserved under some of these products. Afterwards, we will explain how to define verbal wreath products between two groups, (these are semi-direct products that resemble the restricted wreath product), and we will show that the Haagerup property, soficity, Connes' embeddability and other metric approximation properties are preserved under taking verbal wreath products. Finally we will give some applications of them.

17:30 - 18:15

#### Isotropy algebras and Fourier analysis for regular inclusions

Ruy Exel (Universidade Federal de Santa Catarina, Brasil)

Given a regular inclusion $$A\subseteq B$$ of C*-algebras, with $$A$$ abelian, we will define the notion of isotropy algebra'' for any given point $$x$$ in the spectrum of $$A.$$ We will then compare this notion with the usual notion of isotropy groups in the context of groupoid C*-algebras, showing that the isotropy algebra relative to the full groupoid C*-algebra is naturally isomorphic to the full group C*-algebra of the corresponding isotropy group. Based on the slightly generalized notion of isotropy module'', we will then introduce a method to analyze the structure of $$B$$ paralleling the classical Fourier analysis. This talk is based on the paper [Characterizing groupoid C*-algebras of non-Hausdorff étale groupoids, arXiv:1901.09683 [math.OA] [v3] Tue, 1 Jun 2021], joint with David Pitts.