Local structure of operator algebras

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 {dollar}Csp*{dollar}-algebras (pro-{dollar}Csp*{dollar}-algebras); among them are the multipliers of the Pedersen ideal of a {dollar}Csp*{dollar}-algebra. We distinguish these as locally compact pro-{dollar}Csp*{dollar}-algebras and give a characterization of all locally compact {dollar}sigma{dollar}-{dollar}Csp*{dollar}-algebras. We show that in the commutative case, the locally compact {dollar}sigma{dollar}-{dollar}Csp*{dollar}-algebras are exactly those which correspond to locally compact Hausdorff topological spaces. Also we characterize these multipliers among the elements affiliated with the corresponding {dollar}Csp*{dollar}-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 {dollar}Csp*{dollar}-algebras of Lie groups. Then we turn to inverse limits of {dollar}Wsp*{dollar}-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.