Nonseparable Calkin Algebras

Ye Zhang, University of New Hampshire, Durham

Abstract

Suppose $m$ is an infinite cardinal. In this dissertation$,$ we study $m$-compact operators and $m$-Calkin algebras$,$ which are generalizations of compact operators and Calkin algebras. In chapter 1$,$ we introduce the background and some elementary theory we need in this thesis. In chapter 2$,$ we extend some classical results about compact operators and Calkin algebras. In chapter 3$,$ firstly$,$ the strong operator topology and weak operator topology are extended to $m$-strong operator topology ($m$-SOT) and $m$-weak operator topology ($m$-WOT)$,$ respectively. Then$,$ $m$-SOT and $m$-WOT continuous linear functionals are studied. Finally$,$ we define a new convergence $m(\tau)$ on $B(H)$ and some properties are characterized. In chapter 4$,$ we study linear functionals that annihilate the $m$-compact operators and extend a theorem of J. Glimm. The second half of chapter $4$ is about $m$-numerical range$,$ by using the new convergence defined in chapter 3$,$ we characterize the elements in the $m$-numerical range of an operator. Chapter 5 is on $m$-normal operators (operators whose image under the quotient map is normal in the $m$-Calkin algebra). We prove a nonseparable version of the famous BDF theorem$,$ that is$,$ every $m$-normal operator can be written as a sum of a normal operator and $m$-compact operator. Also$,$ we generalize the Weyl-von Neumann-Berg theorem. In chapter 6$,$ we investigate the ultraproducts of $M_{n}(\mathbb{C})$'s. We give characterizations for compact operators and Fredholm operators in these ultraproducts$,$ which are different from the classical cases. Also$,$ we study the maximal chains of projections in the ultraproduct algebra. It is proved that all maximal chains of projections in the ultraproduct algebra$,$ modulo the compact operators$,$ are order isomorphic if we assume the continuum hypothesis.