Local Structure of Operator Algebras

Abstract

In this thesis some aspects of a local theory for operator algebras are explored. The main purpose is to provide some tools for studying locally compact quantum groups. We first consider inverse limits of $C\sp*$-algebras (pro-$C\sp*$-algebras); among them are the multipliers of the Pedersen ideal of a $C\sp*$-algebra. We distinguish these as locally compact pro-$C\sp*$-algebras and give a characterization of all locally compact $\sigma$-$C\sp*$-algebras. We show that in the commutative case, the locally compact $\sigma$-$C\sp*$-algebras are exactly those which correspond to locally compact Hausdorff topological spaces. Also we characterize these multipliers among the elements affiliated with the corresponding $C\sp*$-algebra. As an application, we prove a version of the generalized Stone's theorem, and apply it to show that certain differential operators are affiliated with the group $C\sp*$-algebras of Lie groups. Then we turn to inverse limits of $W\sp*$-algebras and use the techniques of non-commutative topology to study the local structure of Kac algebras. Also we study inverse limits of Kac algebras.