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Quasi-measures and dimension theory. (English) Zbl 0842.28005
A normal space $$X$$ is said to be an $$A$$-space (resp. $$MA$$-space) if every quasi-linear (multiplicative quasi-linear) functional on $$C_b(X)$$, the space of bounded continuous real-valued functions on $$X$$, is linear. The purpose of this paper is to give a sufficient condition for a normal space $$X$$ to be an $$A$$-space or an $$MA$$-space in terms of classical notions of dimension theory such as the large inductive dimension $$\text{Ind}(X)$$ or the Lebesgue covering dimension $$\dim(X)$$. Concerning this, the author shows that if $$X$$ is a normal space and $$\text{Ind}(X)\leq 1$$ (resp. $$\dim(X)\leq 1$$), then $$X$$ is an $$A$$-space ($$MA$$-space). In other words, it is shown that for normal spaces $$X$$ with $$\text{Ind}(X)\leq 1$$ (resp. $$\dim(X)\leq 1$$), every quasi-measure ($$\{0, 1\}$$-valued quasi-measure) on $$X$$ admits a finitely additive extension to the smallest algebra of subsets of $$X$$ which contains the closed sets.

MSC:
 28C15 Set functions and measures on topological spaces (regularity of measures, etc.) 54F45 Dimension theory in general topology
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References:
 [1] Aarnes, J., Physical states on a $$C\^{}\{∗\}- algebra$$, Acta math., 122, 161-172, (1969) · Zbl 0183.14203 [2] Aarnes, J., Quasi-states on $$C\^{}\{∗\}- algebras$$, Trans. amer. math. soc., 149, 601-625, (1970) · Zbl 0212.15403 [3] Aarnes, J., Quasi-states and quasi-measures, Adv. math., 86, 41-67, (1991) · Zbl 0744.46052 [4] Aarnes, J., Pure quasi-states and extremal quasi-measures, Math. ann., 295, 575-588, (1993) · Zbl 0791.46028 [5] Aarnes, J., Construction of non-subadditive measures and discretization of Borel measures, () · Zbl 0842.28004 [6] Akemann, C.; Newberger, S., Physical states on a $$C\^{}\{∗\}- algebra$$, (), 500 · Zbl 0272.46037 [7] Bachman, G.; Sultan, A., On regular extensions of measures, Pacific J. math., 86, 389-395, (1980) · Zbl 0441.28003 [8] Boardman, J., Quasi-measures on completely regular spaces, () · Zbl 0913.46042 [9] Engelking, R.; Sieklucki, K., Topology: A geometrie approach, (1992), Heldermann Verlag Berlin [10] Gillman, L.; Jerison, M., Rings of continuous functions, (1974), Springer New York · Zbl 0151.30003 [11] Halmos, P., Measure theory, (1974), Springer New York [12] F. Knudsen, Topology and the construction of extreme quasi-measures, Adv. Math., to appear. · Zbl 0914.28010 [13] Pears, A., Dimension theory of general spaces, (1975), Cambridge University Press London · Zbl 0312.54001 [14] Wheeler, R., A survey of Baire measures and strict topologies, Exposition. math., 2, 97-190, (1983) · Zbl 0522.28009
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