semigroups of contractions on Hilbert space

I have the following question:

Let $S$ be a convex semigroup - not a priori unital and not necessarily self-adjoint - of contractins on a Hilbert space $H$. Are there any (non-obvious) conditions on $S$ that imply that the identity operator on $H$ is contained in the weak operator closure of $S$?

Any pertinent hints will be appreciated!

Volker Runde.

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