I remember my professor telling us a statemen like:
Every $C_0$-Semigroup on $L^\infty$ admits a bounded Generator.
I currently need a result o this form, so i wonder if this ture. I couldn't find anything about this neither on google nor in my books. Do you remember a statement similiar to this?
As proved by Heinrich P. Lotz, every strongly continuous semigroup of operators on a Grothendieck space with the Dunford-Pettis property is indeed uniformly continuous.
As $L^\infty$ is a Grothendieck space with the Dunford-Pettis property, the result is true.
The relevant definitions, the proof and examples of other spaces in this class can be found in this book.