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Locally Compact Pro-C*-Algebras

Published online by Cambridge University Press:  20 November 2018

Massoud Amini
Massoud Amini*
Affiliation:
Department of Mathematics, Tarbiat Modarres University, P.O. Box 14115-175, Tehran, Iran e-mail: [email protected]
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Abstract

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Let $X$ be a locally compact non-compact Hausdorff topological space. Consider the algebras $C\left( X \right),{{C}_{b}}\left( X \right),{{C}_{0}}\left( X \right),\,\text{and}\,{{C}_{00}}\left( X \right)$ 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-${{C}^{*}}$- 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 ${{C}^{*}}$-algebra ${{C}_{0}}\left( X \right)$, one can get the other three algebras by ${{C}_{00}}\left( X \right)=K\left( {{C}_{0}}\left( X \right) \right),{{C}_{b}}\left( X \right)=M\left( {{C}_{0}}\left( X \right) \right),C\left( X \right)=\Gamma \left( K\left( {{C}_{0}}\left( X \right) \right) \right)$, where the right hand sides are the Pedersen ideal, the multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of ${{C}_{0}}\left( X \right)$, respectively. In this article we consider the possibility of these transitions for general ${{C}^{*}}$-algebras. The difficult part is to start with a pro-${{C}^{*}}$-algebra A and to construct a ${{C}^{*}}$-algebra ${{A}_{0}}$ such that $A=\Gamma \left( K\left( {{A}_{0}} \right) \right)$. The pro-${{C}^{*}}$-algebras for which this is possible are called locally compact and we have characterized them using a concept similar to that of an approximate identity.

Keywords

46L0546M40pro-C* -algebrasprojective limitmultipliers of Pedersen's ideal
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 1 , 01 February 2004 , pp. 3 - 22
Copyright
Copyright © Canadian Mathematical Society 2004

References

[APT] Akemann, C. A., Pedersen, G. K. and Tomiyama, J., Multipliers of C*-algebras. J. Funct. Anal. (3) 13(1973), 277–301.Google Scholar
[CD] Collins, H. S. and Dorroh, J. R., Remarks on certain function spaces. Math. Ann. 176(1968), 157–168.Google Scholar
[CF] Collins, H. S. and Fontenot, R. A., Approximate identities and the strict topology. Pacific J. Math. 43(1972), 63–80.Google Scholar
[FW] Fontenot, R. A. and Wheeler, R. F., Approximate identities and paracompactness. Proc. Amer. Math. Soc. 99(1987), 232–236.Google Scholar
[GJ] Gillman, L. and Jerison, M., Rings of continuous functions. Springer Verlag, New York, Heidelberg, Berlin, 1976.Google Scholar
[Lin] Lin, H. X., Support algebras of σ-unital C*-algebras and their quasi-multipliers. Trans. Amer. Math. Soc. 325(1991), 829–854.Google Scholar
[LT] Lazar, A. and Taylor, D. C., Multipliers of Pedersen's ideal. Mem. Amer. Math. Soc. 169(1976).Google Scholar
[Mur] Murphy, G. J., C*-algebras and operator theory. Academic Press, New York, 1990.Google Scholar
[Pd66] Pedersen, Gert K., Measure theory for C*-algebras I–IV. Math. Scand. 19(1966), 131–145; 22(1968), 63–74; 25(1969), 71–93; 25(1969), 121–127.Google Scholar
[Pd64] Pedersen, Gert K., Measure theory for C*-algebras. Ph.D. Thesis.Google Scholar
[Ph88a] Phillips, N. C., Inverse limits of C*-algebras. J. Operator Theory 19(1988), 159–195.Google Scholar
[Ph88b] Phillips, N. C., A new approach to the multipliers of Pedersen's ideal. Proc. Amer. Math. Soc. (3) 104(1988), 861–867.Google Scholar
[Ph88c] Phillips, N. C., Inverse limits of C*-algebras and applications. Operator Algebras and Applications 1, 127–185, London Math. Soc. Lecture Note Ser. 135, Cambridge Univ. Press, Cambridge, New York, 1988.Google Scholar
[Ty72] Taylor, D. C., A general Phillips theorem for C*-algebras and some applications. Pacific J. Math. 40(1972), 477–488.Google Scholar
[Whe] Wheeler, R. F., Well-behaved and totally bounded approximate identities for C0(X) . Pacific J. Math. 65(1976), 261–269.Google Scholar