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1. CJM 2004 (vol 56 pp. 3)
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 $X$.
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.
Keywords:pro-$C^\ast$-algebras, projective limit, multipliers of Pedersen's ideal Categories:46L05, 46M40 |