On the square root of a positive B(*)-valued function

Let ([Omega], , [mu]) be a measure space, a separable Banach space, and * the space of all bounded conjugate linear functionals on . Let f be a weak* summable positive B(*)-valued function defined on [Omega]. The existence of a separable Hilbert space , a weakly measurable B()-valued function Q satisfying the relation Q*([omega])Q([omega]) = f([omega]) is proved. This result is used to define the Hilbert space L2,f of square integrable operator-valued functions with respect to f. It is shown that for B+(*)-valued measures, the concepts of weak*, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.