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Paper   IPM / P / 7158
School of Physics
  Title:   Locally Compact Pro-C*-Algebras
  Author(s):  M. Amini
  Status:   Preprint
  Journal:
  No.:  0
  Year:  1999
  Supported by:  IPM
  Abstract:
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:proC*algebras, Pedersen's ideal, multiplier algebra.

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