Quantum law of large numbers for Banach spaces

We consider random operators $\mathrm{\Omega }\to \mathcal{L}({\mathrm{\ell }}_p,{\mathrm{\ell }}_p)$ for some $1⩽p<\mathrm{\infty }$. The law of large numbers is known in the case $p=2$ in the form of usual law of large numbers. Instead of sum of i.i.d. variables there may be considered the composition of random semigroups $e^{A_{i}t\mathrm{/}n}$. We obtain the law of large numbers for the case $p⩽2$.