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 ∗-algebras . We investigate the relation of these ideas to the Akemann-Giles-Kummer non commu...

For a Banach algebra $A$, $A''$ is $(-1)$-Weakly amenable if $A'$ is a Banach $A''$-bimodule and $H^1(A'',A')={0}$. In this paper, among other things,  we study the relationships between the $(-1)$-Weakly amenability of $A''$ and the weak amenability of $A''$ or $A$. Moreover, we show that the second dual of every $C^ast$-algebra is $(-1)$-Weakly amenable.

we investigate the problem of the existence of a frame forright ideals of a c*-algebra a, without the use of the kasparov stabilizationtheorem. we show that this property can not characterize a as a c*-algebraof compact operators.