Measures on Projections in a $W^*$-Algebra of Type $I_2$

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.