A new approach to $\gamma$-bounded representations

Let $X$ be a Banach space, let $(\Omega,\mu)$ be a $\sigma$-finite measure space and let $A,B\colon\Omega\to B(X)$ be strongly measurable $\gamma$-bounded functions. We show that for all $x\in X$ and all $x^*\in X^*$, there exist a Hilbert space $K$ and two measurable functions $a_1\in L^\infty(\Omega;K)$ and $a_2\in L^\infty(\Omega;K)$ such that $\langle B(t)A(s)x,x^*\rangle = (a_2(t)\,\vert\, a_1(s))_{K}$ for a.e. $(s,t)$ in $\Omega^2$, with $\Vert a_1\Vert_\infty \Vert a_2\Vert_\infty\leq \gamma(A)\gamma(B)\Vert x\vert\vert x^*\Vert$. This factorization property allows us to improve or simplify some results concerning $\gamma$-bounded representations of groups or semigroups.