Let $G$ and $H$ be locally compact (Hausdorff, second countable) groupoids with Haar systems $\mu$ and $\nu$, respectively. Is it true then that the (full) groupoid $C^*$-algebras satisfy $$ C^*(G\times H, \mu \times \nu) \cong C^*(G,\mu) \otimes C^*(H,\nu), $$ where $\otimes$ is some $C^*$-algebraic tensor product (maybe the maximal one?)? This fact is hinted at, for example, here.

Does anyone know of a reference which contains a proof, or a similar statement under some more restrictive assumptions? I would also be glad for a reference for the group case.


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