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Suppose we have an (abelian) semigroup $S$ acting by endomorphisms on a $C^*$-algebra A giving rise to a semigroup crossed product $B = A\rtimes S$. Are they nice criteria known which ensure $B$ to be nuclear?
I am most interested in the case where $S$ is abelian and $A$ is abelian and unital.
Of course, when $S$ is actually a group then the case I'm interested in is well known to be nuclear, but because in general sub $C^*$-algebras of nuclear ones don't have to be nuclear, one has to be a little bit careful.