On a vector-valued local ergodic theorem in ${\displaystyle L_{\infty }}$

Let ${\displaystyle T=\{T\left(u\right):u\in {ℝ}_d^{+}\}}$ be a strongly continuous d-dimensional semigroup of linear contractions on ${\displaystyle L_1((\Omega ,\Sigma ,\mu );X)}$, where (Ω,Σ,μ) is a σ-finite measure space and X is a reflexive Banach space. Since ${\displaystyle L_1((\Omega ,\Sigma ,\mu );X)\cdot =L_{\infty }((\Omega ,\Sigma ,\mu );X\cdot )}$, the adjoint semigroup ${\displaystyle T\cdot =\{T\cdot \left(u\right):u\in {ℝ}_d^{+}\}}$ becomes a weak*-continuous semigroup of linear contractions acting on ${\displaystyle L_{\infty }((\Omega ,\Sigma ,\mu );X\cdot )}$. In this paper the local ergodic theorem is studied for the adjoint semigroup T*. Assuming that each T(u), ${\displaystyle u\in {ℝ}_d^{+}}$, has a contraction majorant P(u) defined on ${\displaystyle L_1((\Omega ,\Sigma ,\mu );ℝ)}$, that is, P(u) is a positive linear contraction on ${\displaystyle L_1((\Omega ,\Sigma ,\mu );ℝ)}$ such that ${\displaystyle \Vert T\left(u\right)f(\omega )\Vert \le P\left(u\right)\Vert f(·)\Vert (\omega )}$ almost everywhere on Ω for every ${\displaystyle ⨍\in L_1((\Omega ,\Sigma ,\mu );X)}$, we prove that the local ergodic theorem holds for T*.