Non Commutative Topology and Local Structure of Operator Algebras

Abstract

Starting with a $W^{\ast }$-algebra $M$ we use the inverse system obtained by cutting down $M$ by its (central) projections to define an inverse limit of $W^{\ast }$-algebras, and show that how this pro-$W^{\ast }$-algebra encodes the local structure of $M$. For the $C^{\ast }$-algebras we do the same thing using their atomic enveloping $W^{\ast }$-algebra . We investigate the relation of these ideas to the Akemann-Giles-Kummer non commutative topology. Finally we use these ideas to look at the local structure of Kac algebras.

23 pages