Locally Compact Pro-C*-Algebras

Abstract

AbstractLet X be a locally compact non-compact Hausdorff topological space. Consider the algebras C(X), Cb(X), C0(X), and C00(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*- algebra). The interesting fact about these algebras is that if one of them is given, the others can be obtained using functional analysis tools. For instance, given the C*-algebra C0(X), one can get the other three algebras by C00(X) = K(C0(X)), Cb(X) = M(C0(X)), C(X) = Γ(K(C0(X))), where the right hand sides are the Pedersen ideal, the multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of C0(X), respectively. In this article we consider the possibility of these transitions for general C*-algebras. The difficult part is to start with a pro-C*-algebra A and to construct a C*-algebra A0 such that A = Γ(K(A0)). The pro-C*-algebras for which this is possible are called locally compact and we have characterized them using a concept similar to that of an approximate identity.