Locally Compact Pro-$C^*$-Algebras

[摘要] Let $X$ be a locally compact non-compact Hausdorff topological space. Considerthe 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 $X$.Of these, the second and third are $C^*$-algebras, the fourth is a normed algebra,whereas the first is only a topological algebra (it is indeed a pro-$C^ast$-algebra).The interesting fact about these algebras is that if one of them is given, theothers can be obtained using functional analysis tools. For instance, given the$C^ast$-algebra $C_0(X)$, one can get the other three algebras by$C_{00}(X)=Kigl(C_0(X)igr)$, $C_b(X)=Migl(C_0(X)igr)$, $C(X)=Gammaigl(K(C_0(X))igr)$, where the right hand sides are the Pedersen ideal, themultiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of$C_0(X)$, respectively. In this article we consider the possibility of thesetransitions for general $C^ast$-algebras. The difficult part is to start with apro-$C^ast$-algebra $A$ and to construct a $C^ast$-algebra $A_0$ such that$A=Gammaigl(K(A_0)igr)$. The pro-$C^ast$-algebras for which this ispossible are called {it locally compact/} and we have characterized them usinga concept similar to that of an approximate identity.

[学科分类] 数学（综合）