Measures on projections in a $W^{\ast }$-algebra of type $I_2$

Abstract

It is shown that for every measure $m$ on projections in a $W^{\ast }$-algebra of type $I_2$, there exists a Hilbert-valued orthogonal vector measure $\mu$ such that $\Vert \mu (p){\Vert }^2=m(p)$ for every projection $p$. With regard to J. Hamhalter's result (Proc. Amer. Math. Soc., 110 (1990), 803-806) it means that the assertion is valid for an arbitrary $W^{\ast }$-algebra.

8 pages