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Fubini and martingale theorems in C* -algebras

Published online by Cambridge University Press:  24 October 2008

C. J. K. Batty
C. J. K. Batty
Affiliation:
Mathematical Institute, Oxford
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Extract

The basic integration theory of Radon measures on locally compact spaces, as described in (5), has been developed in various directions both in commutative and non-commutative analysis. Thus if Ω is a compact Hausdorff space, and C(Ω) denotes the space of continuous complex-valued functions on Ω, Kaplan (16) showed how topological measure theory can be performed in the second dual C(Ω)** of C(Ω). Here as usual C(Ω)* is identified with the space of Radon measures on Ω by associating with a measure μ the linear functional φμ where

Thus if f is a bounded real-valued function on Ω, there is an affine function f* defined on the set P(Ω) of Radon probability measures μ on Ω, by

and f* extends by linearity to a functional in C(Ω)**. Conversely any function x on P(Ω) determines a function x0 on Ω. by

wehere εω is the unit point of mass at ω.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 1 , July 1979 , pp. 57 - 69
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCES

(1)Akemann, C. A. and Pedersen, G. K.Complications of semi-continuity in c*-algebra theory. Duke Math. J. 40 (1973), 785–795.CrossRefGoogle Scholar
(2)Alfsen, E. M.Compact convex sets and boundary integrals (Berlin, Springer, 1971).Google Scholar
(3)Alo, R. A., Cheney, C. A. and de Korvin, A.Martingales in Banach *-algebras. J. Reine Angew. Math. 273 (1975), 49–56.Google Scholar
(4)Sparre Andersen, E. and Jessen, B.Some limit theorems on integrals in an abstract set. Danske Vid. Selsk. Mat.-Fys. Medd. 22 (1946), no. 14.Google Scholar
(5)Bourbaki, N.Intégration, 2nd ed., chaps. I–IV (Paris, Hermann, 1965).Google Scholar
(6)Combes, F.Quelques propriétés des C*-algèbres. Bull. Sci. Math. (2) 94 (1970), 165–192.Google Scholar
(7)Cuculescu, I.Martingales on von Neumann algebras. J. Multivariate Analysis 1 (1971), 17–27.CrossRefGoogle Scholar
(8)Dang-Ngoc, N. Pointwise convergence of martingales in von Neumann algebras. (To appear.)Google Scholar
(9)Davies, E. B.On the Borel structure of C*-algebras. Comm. Math. Phys. 8 (1968), 147–163.Google Scholar
(10)Davies, E. B.The structure of ∑*-algebras. Quart. J. Math. (Oxford) (2) 20 (1969), 351–366.Google Scholar
(11)Dixmier, J.Les algèbres des opérateurs dans l'espace hilbertien, 2nd ed. (Paris, Gauthier–Villars, 1969).Google Scholar
(12)Doob, J. L.Stochastic processes (New York, Wiley, 1953).Google Scholar
(13)Dunford, N. and Schwartz, J. T.Linear operators (New York, Interscience, 1958).Google Scholar
(14)Grothendieck, A.Produits tensoriels et espaces nucléaires. Mem. Amer. Math. Soc. 16 (1955).Google Scholar
(15)Kadison, R. V.Unitary invariants for representation of operator algebras. Ann. Math. 66 (1957), 304–379.Google Scholar
(16)Kaplan, S.On the second dual of the space of continuous functions. Trans. Amer. Math. Soc. 86 (1957), 70–90.CrossRefGoogle Scholar
(17)Lance, E. C.Martingale convergence in von Neumann algebras. Math. Proc. Cambridge Philos. Soc. 84 (1978), 47–56.CrossRefGoogle Scholar
(18)Namioka, I. and Phelps, R. R.Tensor products of compact convex sets. Pacific J. Math. 31 (1969), 469–480.CrossRefGoogle Scholar
(19)Pedersen, G. K.Measure theory for C*-algebras III. Math. Scand. 25 (1969), 71–93.CrossRefGoogle Scholar
(20)Pedersen, G. K.On weak and monotone σ-closures of C*-algebras. Comm. Math. Phys. 11 (1969), 221–226.Google Scholar
(21)Pedersen, G. K.Applications of weak* semi-continuity in C*-algebra theory. Duke Math. J. 39 (1972), 431–450.Google Scholar
(22)Plymen, R. J.C*-algebras and Mackey's axioms. Comm. Math. Phys. 8 (1968), 132–146.CrossRefGoogle Scholar
(23)Sakai, S.C*-algebras and W*-algebras (Berlin, Springer, 1971).Google Scholar
(24)Schatten, R.A theory of cross-spaces (Princeton, N.J., Princeton University Press, 1950).Google Scholar
(25)Tomiyama, J.On the projection of norm one in W*-algebras, Proc. Japan Acad. 33 (1957), 608–612.Google Scholar
(26)Tomiyama, J.Applications of Fuibini type theorem to the tensor product of C*-algebras. Tôhoku Math. J. (2) 19 (1967), 213–226.CrossRefGoogle Scholar
(27)Umegaki, H.Conditional expectations in an operator algebra II. Tôhoku Math. J. (2) 8 (1956), 86–99.Google Scholar