The Space of Operator Valued Functions Seen as Hilbert $H^*$-Module

Zlatko Lazović

Let $M$ be a space of weakly$^*$-measurable functions $\mathcal F\colon\Omega\to B(H)$ on measure space $(\Omega,\Sigma,\mu)$, for which the function $\mathcal F^*\mathcal F$ is Gel'fand integrable and Gel'fand integral $\int_{\Omega}\mathcal F^*\mathcal F\,d\mu$ is a nuclear operator on Hilbert space $H$. We show that $M$ is Hilbert $H^*$-module which contains an orthonormal basis.