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Paper IPM / P / 7158  


Abstract:  
Let X be a locally compact non compact Hausdorff
topological space. Consider the algebras C(X),C_{b}(X), C_{0}(X), 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) = Γ(K(C_{0}(X))). Also each algebra in the above
list can be obtained from the previous one as follows:
C_{0}(X)=C^{*}−completion of C_{00}(X),C_{b}(X)=b(C(X))=elements with bounded spectrum,
and, if X is second countable,
C_{0}(X)={f ϵC_{b}(X):fC_{b}(X) separable} [Wr95]. this article we consider the possibility of these transitions
for general C^{*}−algebras. We use the same notation as
in the classical case to distinguish the objects of each category.
Therefore if a C^{*}−algebra is denoted by A_{0}, then
its Pedersen's ideal is denoted by A_{00}, and the multiplier
algebra of A and A_{00} are denoted
by A_{b} and A respectively. textbfKeywords:pro−C^{*}−algebras, Pedersen's ideal, multiplier algebra. Download TeX format 

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