This question is related to this question [link](https://mathoverflow.net/questions/61818/about-properties-of-groupoid-c-algebras).


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.


Thanks!



Nuclearity of certain semigroup crossed product C*-algebras
At least in the case that $S$ is the positive integers, this is discussed in the paper by G. Murphy, "Crossed products of $C^\ast$-algebras by endomorphisms", Int. Eq. and Operator Th. Volume 24, Number 3, 298-319, DOI: 10.1007/BF01204603. His result is that the crossed product is nuclear iff $A$ is nuclear. 


