A mean ergodic theorem of an amenable group action

We consider a sequence of weak Kadison-Schwarz maps T-n on a von-Neumann algebra M with a faithful normal state phi sub-invariant for each (T-n, n = 1) and use a duality argument to prove strong convergence of their pre-dual maps when their induced contractive maps (T-n, n = 1) on the GNS space of (M, phi) are strongly convergent. The result is applied to deduce improvements of some known ergodic theorems and Birkhoff's mean ergodic theorem for any locally compact second countable amenable group action on the pre-dual Banach space M-*.