Locally Compact Pro-$C^{*}$-Algebras

Let $X$ be a locally compact non compact Hausdorff topological space. Consider the algebras $C(X)$, $C_b(X)$, $C_0(X)$, and $C_{00}(X)$ of respectively arbitrary, bounded, vanishing at infinity, and compactly supported continuous functions on $X$. From these, the second and third are $C^{*}$-algebras, the forth is a normed algebra, where as the first is only a topological algebra. The interesting fact about these algebras is that if one of them is given, the rest can be obtained using functional analysis tools. For instance, given the $C^{*}$-algebra $C_0(X)$, one can get the other three algebras by $C_{00}(X)=K(C_0(X))$, $C_b(X)=M(C_0(X))$, $C(X)=\Gamma(K(C_0(X)))$, that is by forming the Pedersen's ideal, the multiplier algebra, and the unbounded multiplier algebra of the Pedersen's ideal, respectively.In this article we consider the possibility of these transitions for general $C^{*}$-algebra . The difficult part is to start with a pro_$C^{*}$-algebra $A$ and to construct a $C^{*}$-algebra $A_0$ such that $A=\Gamma(K(A_0))$. The pro-$C^{*}$-algebras for which this is possible are called {\it locally compact} and we have characterized them using a concept similar to approximate identities.