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| Paper IPM / Physic / 7158 |
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| Abstract: | |
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Let X be a locally compact non compact Hausdorff
topological space. Consider the algebras C(X),Cb(X), C0(X), C00(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
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))). Also each algebra in the above
list can be obtained from the previous one as follows:
C0(X)=C*−completion of C00(X),Cb(X)=b(C(X))=elements with bounded spectrum,
and, if X is second countable,
C0(X)={f ϵCb(X):fCb(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 A0, then
its Pedersen's ideal is denoted by A00, and the multiplier
algebra of A and A00 are denoted
by Ab and A respectively. textbfKeywords:pro−C*−algebras, Pedersen's ideal, multiplier algebra. Download TeX format |
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