On the Adjoint of a Strongly Continuous Semigroup

Abstract

Using some techniques from vector integration, we prove the weak measurability of the adjoint of strongly continuous semigroups which factor through Banach spaces without isomorphic copy of $l_1$; we also prove the strong continuity away from zero of the adjoint if the semigroup factors through Grothendieck spaces. These results are used, in particular, to characterize the space of strong continuity of ${\left\{T^{**}\left(t\right)\right\}}_{t\ge 0}$, which, in addition, is also characterized for abstract $L$- and $M$-spaces. As a corollary, it is proven that abstract $L$-spaces with no copy of $l_1$ are finite-dimensional.