Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

This question is related to this question link.

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!

share|improve this question
2  
If one is happy to use the (deep) equivalence of nuclearity and amenability for C*-algebras, then Theorem 3 of Rosenberg's paper "Amenability of crossed products of C*-algebras" (Comm Math Phys 1977) has some results, at least when $S$ is the positive integers. –  Yemon Choi Apr 15 '11 at 19:07
add comment

1 Answer

up vote 4 down vote accepted

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.

share|improve this answer
    
Thank you very much for this reference, which covers pretty much the abelian cases in general (not only the positive integers). –  user5831 Apr 15 '11 at 19:42
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.