Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Approximate identities and paracompactness


Authors: R. A. Fontenot and R. F. Wheeler
Journal: Proc. Amer. Math. Soc. 99 (1987), 232-236
MSC: Primary 46J10; Secondary 46E25, 54D18
DOI: https://doi.org/10.1090/S0002-9939-1987-0870777-3
MathSciNet review: 870777
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ denote a locally compact Hausdorff space and $ {C_b}(X)$ the algebra of continuous complex-valued functions on $ X$. The main result of this paper is that $ X$ is paracompact if and only if $ {C_0}(X)$, the subalgebra of $ {C_b}(X)$ consisting of functions which vanish at infinity, has an approximate identity which is a relatively compact subset of $ {C_b}(X)$ for the weak topology of the pairing of $ {C_b}(X)$ with its strict topology dual.


References [Enhancements On Off] (What's this?)

  • [1] R. C. Buck, Bounded continuous functions on a locally compact space, Michigan Math. J. 5 (1958), 95-104. MR 0105611 (21:4350)
  • [2] H. S. Collins, Strict, weighted, and mixed topologies and applications, Adv. in Math. 19 (1976), 207-237. MR 0394149 (52:14953)
  • [3] H. S. Collins and J. R. Dorroh, Remarks on certain function spaces, Math. Ann. 176 (1968), 157-168. MR 0222644 (36:5694)
  • [4] H. S. Collins and R. A. Fontenot, Approximate identities and the strict topology, Pacific J. Math. 43 (1972), 63-80. MR 0313824 (47:2378)
  • [5] J. B. Conway, The strict topology and compactness in the space of measures, Trans. Amer. Math. Soc. 126 (1967), 474-486. MR 0206685 (34:6503)
  • [6] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
  • [7] N. Dunford and J. T. Schwartz, Linear operators, Part I: General theory, Interscience, New York, 1958. MR 1009162 (90g:47001a)
  • [8] A. Grothendieck, Critères de compacité dans les espaces fonctionnels generaux, Amer. J. Math. 74 (1952), 168-186. MR 0047313 (13:857e)
  • [9 J] D. Pryce, A device of R. J. Whitley's applied to pointwise compactness in spaces of continuous functions, Proc. London Math. Soc. (3) 23 (1971), 532-546. MR 0296670 (45:5729)
  • [10] I. Z. Ruzsa, personal communication, 1974.
  • [11] F. D. Sentilles, Bounded continuous functions on a completely regular space, Trans. Amer. Math. Soc. 168 (1972), 311-336. MR 0295065 (45:4133)
  • [12] D. C. Taylor, A general Phillips theorem for $ {C^ * }$-algebras and some applications, Pacific J. Math. 40 (1972), 477-488. MR 0308799 (46:7913)
  • [13] R. F. Wheeler, Well-behaved and totally bounded approximate identities for $ {C_0}(X)$, Pacific J. Math. 65 (1976), 261-269. MR 0458150 (56:16353)
  • [14] -, A survey of Baire measures and strict topologies, Exposition Math. 2 (1983), 97-190. MR 710569 (85b:46035)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46J10, 46E25, 54D18

Retrieve articles in all journals with MSC: 46J10, 46E25, 54D18


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0870777-3
Keywords: Approximate identity, paracompact, weakly compact
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society