  Starting with a $W^{*}$-algebra $M$ we use the inverse system obtained by
cutting down $M$ by its (central) projections to define an inverse limit of
$W^{*}$-algebras, and show that how this pro-$W^{*}$-algebra encodes the local
structure of $M$. For the $C^{*}$-algebras we do the same thing using their
atomic enveloping $W^{*}$-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.
