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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.


  • 2
    $\begingroup$ 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. $\endgroup$
    – Yemon Choi
    Apr 15 '11 at 19:07

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.

  • $\begingroup$ Thank you very much for this reference, which covers pretty much the abelian cases in general (not only the positive integers). $\endgroup$
    – user5831
    Apr 15 '11 at 19:42

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