Global Central Limit Theorems for Markov chains with normal transition operator

Let $P$ be a Markov operator on a general state space $(S,Σ)$ with invariant probability $m$, assumed ergodic. When $P$ is normal in $L_2(m)$, we prove the necessity of uniform ergodicity, under some spectral conditions (satisfied when $P$ is symmetric), for the annealed CLT to hold for {\it every}