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)



Signed quasi-measures and dimension theory

Author: D. J. Grubb
Journal: Proc. Amer. Math. Soc. 128 (2000), 1105-1108
MSC (1991): Primary 28C15
Published electronically: August 5, 1999
MathSciNet review: 1636950
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A quasi-linear functional on $C(X)$ is a real-valued function that is linear on each closed, singly generated subalgebra and is norm bounded. We show that if the covering dimension $\dim X\leq 1$, then every quasi-linear functional on $C(X)$ is, in fact, linear. We do this by considering an associated set function, called a quasi-measure, and ask when such a set function can be extended to be a measure.

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

  • 1. Johan Aarnes, Quasi-states and quasi-measures, Adv. in Math. 86, (1991) pp. 41-67. MR 92d:46152
  • 2. L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York, 1960. MR 22:6994
  • 3. D. J. Grubb, Signed Quasi-Measures, Trans. Amer. Math. Soc. 349, (1997) pp. 1081-1089. MR 98c:28012
  • 4. D. J. Grubb and Tim LaBerge, Spaces of quasi-measures, accepted Bull. Math. Canad.
  • 5. A. R. Pears, Dimension Theory of General Spaces, Cambridge University Press, London 1975. MR 52:15405
  • 6. K. P. S. Rao and M. B. Rao, Theory of Charges, Academic Press, New York, 1983. MR 86f:28006
  • 7. Dmitri Shakmatov, Linearity of quasi-states on commutative $C^*$ algebras of stable rank 1. unpublished.
  • 8. Robert Wheeler, Quasi-Measures and Dimension Theory, Top. Appl. 66, (1995) pp. 75-92. MR 96m:28002

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 28C15

Retrieve articles in all journals with MSC (1991): 28C15

Additional Information

D. J. Grubb
Affiliation: Department of Mathematical Sciences, Northern Illinois University, De Kalb, Illinois 60115

Received by editor(s): February 10, 1998
Received by editor(s) in revised form: June 1, 1998
Published electronically: August 5, 1999
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society