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In commutative case, it does hold true: ${\rm dim}(E[H])=1\iff \phi$ is ergodic. Maybe, it is true also if the support of $\phi$ in the bidual is central.
No Adrian, ergodicity means extremality among invariant states. ${\rm dim}(E[H])=1$ implies ergodicity, but the converse is true under the additional assumption of $G$-abelianess (in our situation $Z$-abelianess), see Prop. 3.1.12 in Sakai's book. The question is that, at my best knowledge, conterexamples don't exist in literature
