If time were a graph, what would evolution equations look like?

Linear evolution equations are considered usually for the time variable being defined on an interval where typically initial conditions or time-periodicity of solutions are required to single out certain solutions. Here we would like to make a point of allowing time to be defined on a metric graph or network where on the branching points coupling conditions are imposed such that time can have ramifications and even loops. This not only generalizes the classical setting and allows for more freedom in the modeling of coupled and interacting systems of evolution equations, but it also provides a unified framework for initial value and time-periodic problems. For these time-graph Cauchy problems questions of well-posedness and regularity of solutions for parabolic problems are studied along with the question of which time-graph Cauchy problems cannot be reduced to an iteratively solvable sequence of Cauchy problems on intervals. Based on two different approaches - an application of the Kalton-Weis theorem on the sum of closed operators and an explicit computation of a Green's function - we present the main well-posedness and regularity results. We further study some qualitative properties of solutions. While we mainly focus on parabolic problems we also explain how other Cauchy problems can be studied along the same lines. This is exemplified by discussing coupled systems with constraints that are non-local in time akin to periodicity.Comment: 30 pages, 4 figure

The Haagerup approximation property for von Neumann algebras via quantum Markov semigroups and Dirichlet forms

The Haagerup approximation property for a von Neumann algebra equipped with a faithful normal state $\varphi$ is shown to imply existence of unital, $\varphi$-preserving and KMS-symmetric approximating maps. This is used to obtain a characterisation of the Haagerup approximation property via quantum Markov semigroups (extending the tracial case result due to Jolissaint and Martin) and further via quantum Dirichlet forms.Comment: 26 pages; v3 adds Corollary 5.8, corrects a mistake in Section 3 and updates references; all the main results remain unchanged. The article will appear in the Communications in Mathematical Physic

Derivations and KMS-Symmetric Quantum Markov Semigroups

We prove that the generator of the $L^2$ implementation of a KMS-symmetric quantum Markov semigroup can be expressed as the square of a derivation with values in a Hilbert bimodule, extending earlier results by Cipriani and Sauvageot for tracially symmetric semigroups and the second-named author for GNS-symmetric semigroups. This result hinges on the introduction of a new completely positive map on the algebra of bounded operators on the GNS Hilbert space. This transformation maps symmetric Markov operators to symmetric Markov operators and is essential to obtain the required inner product on the Hilbert bimodule.Comment: 34 page

Degenerate C-ultradistribution semigroups in locally convex spaces

The main subject in this paper are degenerate C-ultradistribution semigroups in barreled sequentially complete locally convex spaces. Here, the regularizing operator C is not necessarily injective and the infinitesimal generator of semigroup is a multivalued linear operator. We also consider exponential degenerate C-ultradistribution semigroups.Bulletin t. 150 de l'Académie serbe des sciences et des arts. Classe des sciences mathématiques et naturelles. Sciences mathematiques no 42

Mini-Workshop: Operator Algebraic Quantum Groups

This mini-workshop brought together a rich and varied cross-section of young and active researchers working on operator algebraic aspects of quantum group theory. The primary goals of this meeting were to highlight the state-of-the-art results on the subject and to trigger new research by advertising some of the main open directions in operator algebraic quantum group theory: classification problems for C$^\ast$- and von Neumann algebras, relations to free/non-commutative probability, applications in quantum information theory, and the creation of new quantum groups and potential classification results for subclasses of quantum groups