Categoria: Operator Algebras

Martin Mathieu (Queen’s University Belfast)

A contribution to Kaplansky’s problem

Abstract: A Jordan homomorphism between two unital, complex algebras A and B is a linear mapping T such that $T(x^2)=(Tx)^2$ for all $x\in A$. Equivalently, T preserves the Jordan product $xy+yx$. Every surjective unital Jordan homomorphism preserves invertible elements. In 1970, Kaplansky asked whether the following converse is true: Suppose $T\colon A\to B$ is a unital surjective invertibility-preserving linear mapping between unital (Jacobson) semisimple Banach algebras A and B. Does it follow that T is a Jordan homomorphism?

In the past 50 years a lot of progress has been made towards a positive solution to Kaplansky’s problem, however, as it stands, it is still open. We will report on some recent joint work with Francois Schulz (University of Johannesburg, SA) which gives a positive answer if B is a C*-algebra with faithful tracial state. Until recently, the existence of traces had been a major obstacle to a solution. Moreover, in our approach, no assumption on the existence of projections (such as real rank zero) is necessary.

I will further discuss a sharpening of Kaplansky’s problem in which the assumption on T is reduced to the preservation of the spectral radius only (a spectral isometry).

Laurent Cantier (Universitat Autònoma de Barcelona)

The Cu$_1$-semigroup as an invariant for K$_1$-obstruction cases

Abstract: The aim of this talk is to explicitly shows that the unitary Cuntz semigroup, defined using the Cuntz semigroup and the K$_1$ group, strictly contains more information than the latter invariants alone. To that end, we construct two C*-algebras, distinguished by their unitary Cuntz semigroup, whose K-Theory and Cu-semigroup are isomorphic. Both A and B, constructed as inductive limits of NCCW 1-algebras, are non-simple unital separable C∗-algebras of stable rank one with K$_1$-obstructions. This shows that a likewise invariant is necessary in order to extend classification results of C*-algebras by means of Cuntz semigroup to the non trivial K$_1$ group case.

Ado Dalla Costa (Universidade Federal de Santa Catalina)

Free actions of groups on separated graphs and their associated C*-algebras

Abstract: I will report on joint work with Alcides Buss and Pere Ara on the study of free actions of groups on separated graphs and explain how this structure reflects on the level of their associated C*-algebras. We prove a version of the Gross-Tucker theorem in this context and show how this can be used to describe the various C*-algebras attached to separated graphs carrying a free action. All this leads to certain Landstad-type duality theorems involving these algebras.

Pere Ara (Universitat Autònoma de Barcelona)

The inverse semigroup of a separated graph

Abstract: For a directed graph $E$, the graph semigroup $S(E)$ was defined by Ash and Hall in 1975. The graph semigroup $S(E)$ is an inverse semigroup, and has been studied by many authors in connection with the theories of graph C*-algebras, Leavitt path algebras, and topological groupoids. For a separated graph $(E,C)$, the direct analogue of $S(E)$ is not an inverse semigroup in general. However, we will introduce an inverse semigroup $IS(E,C)$ for each separated graph, which produces the same graph semigroup $S(E)$ as above in the non-separated case. We will develop a normal form of the elements of $IS(E,C)$ in close analogy to the Scheiblich normal form for elements of the free inverse semigroup.

This is joint work in progress with Alcides Buss and Ado Dalla Costa, both from Universidade Federal de Santa Catarina (Brazil).

This is a seminar online that ran from the end of November 2021 until mid 2022. It was organized together with Ramon Antoine and Hannes Thiel.

Details of the talks programmed may be found here.

Eduard Ortega (NTNU Trondheim)

Left cancellative small categories and their associated algebras

Abstract: In this talk I will explain how to associate an étale groupoid to a left cancellative small category. We will show that certain categories with a length function can be written as a Zappa-Zsép product of a free subcategory and the groupoid of invertible elements. This talk is based in a common project with Enrique Pardo.