Disjointness of C-dynamical systems

We study ergodic theorems for disjoint C*- and W*-dynamical systems, where disjointness here is a noncommutative version of the concept introduced by Furstenberg for classical dynamical systems. We also consider speci c examples of disjoint W*-dynamical systems. Lastly we use unique ergodicity and unique ergodicity relative to the xed point algebra to give examples of disjoint C*-dynamical systems.The National Research Foundation of South Africahttp://www.math.uh.edu/~hjm/am2016Mathematics and Applied MathematicsPhysic

Geographical and temporal distribution of SARS-CoV-2 clades in the WHO European Region, January to June 2020

We show the distribution of severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) genetic clades over time and between countries and outline potential genomic surveillance objectives. We applied three genomic nomenclature systems to all sequence data from the World Health Organization European Region available until 10 July 2020. We highlight the importance of real-time sequencing and data dissemination in a pandemic situation, compare the nomenclatures and lay a foundation for future European genomic surveillance of SARS-CoV-2

Geographical and temporal distribution of SARS-CoV-2 clades in the WHO European Region, January to June 2020

We show the distribution of SARS-CoV-2 genetic clades over time and between countries and outline potential genomic surveillance objectives. We applied three available genomic nomenclature systems for SARS-CoV-2 to all sequence data from the WHO European Region available during the COVID-19 pandemic until 10 July 2020. We highlight the importance of real-time sequencing and data dissemination in a pandemic situation. We provide a comparison of the nomenclatures and lay a foundation for future European genomic surveillance of SARS-CoV-2.Peer reviewe

Closed two-sided ideals in a von Neumann algebra and applications

The aim of this thesis is to study closed two-sided ideals in a von Neumann algebra A, not only by looking into the structure of these ideals, but by using them in several applications on the theory of von Neumann algebras. For example, one of the main objects of this thesis is to develop a Riesz theory relative to any closed ideal in a von Neumann algebra by proving some characterization theorems of relatively Riesz operators and then to use this to prove a Riesz decomposition theorem. Section 1 contains the definitions of some basic facts concerning von Neumann algebras used throughout this work. The main issue of section 2 is to consider three specific examples of closed two-sided ideals in a semifinite algebra with a non-zero type I direct summand, namely the ideals of operators compact relative to the von Neumann algebra, the ideal of compact operators contained rn A and the ideal of the so called Rosenthal operators relative to A. These ideals are used to obtain factorization results as well as a duality theorem. In the third section we deduce geometrical characterizations as well as a spectral characterization for the quotient norm on A/1, where 1 is any closed ideal in A. We then prove some characterization theorems on the semi-Fredholm elements relative to 1. In section 4 Riesz operators relative to a closed two-sided ideal are defined. The results in this section are similar to those known for the classical case and they are used in the sequel to prove characterization theorems for relatively Riesz operators as well as a Riesz decomposition theorem. In section 5 a geometrical characterization of Riesz operators relative to any closed ideal is proved. This geometrical characterization is used in section 6 to obtain a Riesz decomposition theorem for Riesz operators relative to specific closed ideals in a semifinite van Neumann algebra.Dissertation (MSc)--University of Pretoria, 1989.Mathematics and Applied MathematicsMScUnrestricte

Fredholm theory in Von Neumann algebras

The main goal of this study is to generalize the theory of compact and of Fredholm operators defined on a complex Hilbert space H to von Neumann algebras. Since this generalization depend heavily on the study of the project ion lattice existing on a von Neumann algebra, the first chapter contains a comprehensive amount of standard material concerning the geometry of projections in a von Neumann algebra A. If we consider the commutant .A.' of a von Neumann algebra and a projection E in .A. then the restriction of each element of .A.' to E(H) defines a representation HE of .A.' into the C* - algebra of all bounded linear operators on E(H) (E(H) is the range space of the projection E). In Chapter 2 we consider all these representations of .A. ' into E ( H) ( where E is assumed to be finite relative to .A.), to construct a commutative monoid M. The Grothendieck group r of M can canonically be equipped with an order relation. This group is important in the Chapters that follow, since it contains the so called indices of the Fredholm elements defined on a von Neumann algebra .A. In Chapter 3 the concept of finite, compact and Fredholm elements are introduced. On the set of all Fredholm elements relative to .A. an index mapping is defined with values in the Grothendieck group r. These values are called the indices of the Fredholm elements relative to .A. The main theorems of this study are obtained in Chapter 4. These results generalize theorems, obtained by F. Riesz and Atkinson.Dissertation (MSc)--University of Pretoria, 1987.Mathematics and Applied MathematicsMScUnrestricte