$\mu$-norm of an operator

Let $({\cal X},\mu)$ be a measure space. For any measurable set $Y\subset{\cal X}$ let $1_Y : {\cal X}\to{\mathbb R}$ be the indicator of $Y$ and let $\pi_Y$ be the orthogonal projector $L^2({\cal X})\ni f\mapsto\pi_Y f = 1_Y f$. For any bounded operator $W$ on $L^2({\cal X},\mu)$ we define its $\mu$-norm $\|W\|_\mu = \inf_\chi\sqrt{\sum \mu(Y_j) \|W\pi_Y\|^2}$, where the infinum is taken over all measurable partitions $\chi = \{Y_1,\ldots,Y_J\}$ of ${\cal X}$. We present some properties of the $\mu$-norm and some computations. Our main motivation is the problem of the construction of a quantum entropy.