Crossed Products of direct sums of $G$-C*-Algebras

Question

Let $G$ be discrete group and $A,B$ two $G$-C*-Algebras. Then $A \oplus B$ inherits a $G$ action by $g(a,b)=(ga,gb)$.

Now, if I take the (reduced) cross product, do I get $$(A\oplus B)\rtimes_{(r)}G \cong A\rtimes{(r)}G \oplus B\rtimes{(r)}G?$$ It seems right, but I am not entirely sure.

If $A=B$, does that change things?

Thank you

Answer 1

This is true. More generally, a similar result is true an arbitrary direct sum: $$\big(\bigoplus A_i\big)\rtimes_{(r)}G\cong \bigoplus_i \big(A_i\rtimes_{(r)}G\big).$$ As indicated above, this holds true for both full and reduced crossed product. Indeed it is true for any ''exotic'' crossed product. See Lemma 2.2 in https://arxiv.org/pdf/1905.07730.pdf