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.

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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/

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