Abstract view
Locally Compact Pro$C^*$Algebras


Published:20040201
Printed: Feb 2004
Abstract
Let $X$ be a locally compact noncompact 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 $X$.
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, the
others 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
$C_{00}(X)=K\bigl(C_0(X)\bigr)$, $C_b(X)=M\bigl(C_0(X)\bigr)$, $C(X)=\Gamma\bigl(
K(C_0(X))\bigr)$, where the right hand sides are the Pedersen ideal, the
multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of
$C_0(X)$, respectively. In this article we consider the possibility of these
transitions for general $C^\ast$algebras. The difficult part is to start with a
pro$C^\ast$algebra $A$ and to construct a $C^\ast$algebra $A_0$ such that
$A=\Gamma\bigl(K(A_0)\bigr)$. The pro$C^\ast$algebras for which this is
possible are called {\it locally compact\/} and we have characterized them using
a concept similar to that of an approximate identity.