Geodesic
Measures on Projections in a $W^*$-Algebra of Type $I_2$
Funkcionalʹnyj analiz i ego priloženiâ, Tome 47 (2013) no. 4, pp. 67-81 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the paper we present two results for measures on projections in a $W^*$-algebra of type $I_2$. First, it is shown that, for any such measure $m$, there exists a Hilbert-valued orthogonal vector measure $\mu$ such that $\|\mu(p)\|^2=m(p)$ for every projection $p$. In view of J. Hamhalter's result (Proc. Amer. Math. Soc., 110 (1990), 803–806), this means that the above assertion is valid for an arbitrary $W^*$-algebra. Secondly, a construction of a product measure on projections in a $W^*$-algebra of type $I_2$ (an analogue of the product measure in classical Lebesgue theory) is proposed.
Keywords: measure on projections, $W^*$-algebra, orthogonal vector measure, product measure. 
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A. N. Sherstnev. Measures on Projections in a $W^*$-Algebra of Type $I_2$. Funkcionalʹnyj analiz i ego priloženiâ, Tome 47 (2013) no. 4, pp. 67-81. http://geodesic.mathdoc.fr/item/FAA_2013_47_4_a5/

[1] J. Hamhalter, “States on $W^*$-algebras and orthogonal vector measures”, Proc. Amer. Math. Soc., 110:3 (1990), 803–806 | MR | Zbl

[2] S. Maeda, “Probability measures on projections in von Neumann algebras”, Rev. Math. Phys., 1:2–3 (1990), 235–290 | MR

[3] A. N. Sherstnev, Metody bilineinykh form v nekommutativnoi teorii mery i integrala, Fizmatlit, M., 2008 | Zbl

[4] S. Sakai, $C^*$-Algebras and $W^*$-Algebras, Springer-Verlag, N.Y.–Heidelberg–Berlin, 1971 | MR | Zbl

[5] I. Segal, “Equivalence of measure spaces”, Amer. J. Math., 73 (1951), 275–313 | DOI | MR | Zbl

[6] A. N. Sherstnev, Measures on projections in a $W^*$-algebra of type $I_2$, arXiv: 1112.5569v1